The Bare Basics
This is going to be a very long post covering much of the info known about Abyssea so far.  If you're new however you might not care to read every single confusing detail.  You probably just want to go in and have some fun with a little advice.  So to start with here's a really quick generalized view of what to expect in Abyssea.

Abyssea blends together elements from Dynamis, Assault, and ZNM in interesting ways.  Abyssea areas are open to anyone with access to the Abyssea addons level 30 and above.  You can enter freely and do as you please however you're limited to the amount of time granted with your visitant status.  Defeating enemies in Abyssea rewards experience and cruor points.  Cruor are points used to buy rewards, buffs, and travel in Abyssea.

To enter an Abyssea area you must purchase the addon then trigger the cutscene in Port Jeuno.  After that speak to the NPC Joachim for your Traverser Stone.  Traverser Stones act similar to an Assault Tag or a Campaign Ops Credit to grant you time in Abyssea (more detailed info later).  Once you have the stone travel to any of the three Cavernous Maws.  All 3 are located in a teleport crag area right before you reach the exit to the starting player areas (e.g. LaTheine maw is right near the exit to West Ronfaure).  Check the Cavernous Maw and choose to enter to continue into the area.  Just be careful not to enter a maw if you have plans to go along on an Abyssea group event within the next hour or you may have to wait (see time restrictions for more info).

Abyssea areas are generally not that hostile to a solo player, unlike Dynamis and Salvage where it's difficult to get around and aggroing means a quick violent death.  All regular enemies in Abyssea can be avoided easily by using sneak and invisible to bypass, or are already non-aggressive.  The areas are very large and so sneaking around isn't really any issue.  Time and slow travel is more of an enemy than accidental aggro.  And so if you are truly unable or unwilling to party up many jobs can find something to do solo in Abyssea.  You could farm some items and cruor from enemies, or do quests as you see fit.  To defeat the more challenging NMs however you will definitely have to party up with a group of friends.

After traveling into Abyssea you'll notice that the areas are nicely distributed with the same types of enemies all neatly clustered together in their own section.  In general, it's pretty simple.  Killing enemies of each type will randomly drop an NM spawn item at a fairly high rate.  Usually that type of enemy will spawn a NM of that same type (e.g. Bat mobs drop trigger for bat NM).  So all you really need to do is find out what NM you're after, travel to those mobs and start farming them until you get an NM trigger.  The trigger target is also conveniently located right where the mobs are.  As you fight in Abyssea the enemies level and spawn rate will tailor itself to your party's.  Eventually the enemies will become Incredibly Tough to your highest members level.  Be careful however, some groups of enemies will not drop any NM triggers because the corresponding NM of their type is already pre-spawned or a lottery.  And so as the enemies become higher level a pre-spawned NM could ultimately become drawn to you as well.

Once you have some NM triggers, it works on a pretty familiar tiered system.  Most NMs require 1 item to spawn, others can require up to 5 triggers.  All of the first tier NM triggers can be purchased on the AH as well.  You will have to work your way up the system to spawn the more powerful NMs after that.

That's basically it.  Go in, target the enemy type of your choice, and get to killing for some exp and cruor.  Figure out which NMs have some nice loot and work your way toward them.  Now if you want to get into the details there's plenty more below.
Rewards
So why should you bother?  What do you get out of it?  There's 3 sets of armor you can purchase with your cruor points directly.  The Perle, Aurore, and Teal sets.  And of course there's all kinds of equipment dropped directly from all the NMs themselves.  As well as the vital rare crafting material
needed for some of the best new synthed body pieces.  There are too many NM items to list here right now and too many of them are undiscovered, so I'll only list cruor reward items and the synthesis materials.  Each item costs 3000 cruor for the shoes, pants, and gloves.  4000 for the hat, and 5000 for the body armor.

78     Perle Salade Rare
[Head] All Races
DEF: 37 STR +5 VIT +5 Attack +7 Haste +2%
Set: Haste +5%
Lv. 78 WAR / PLD / DRK / BST / SAM / DRG
78     Perle Hauberk Rare

[Body] All Races
DEF: 61 STR +8 DEX +4 Accuracy +8 Attack+8
"Critical hit rate" +2% Set: Haste +5%
Lv. 78 WAR / PLD / DRK / BST / SAM / DRG
78     Perle Moufles Rare

[Hands] All Races
DEF: 23 STR +4 DEX +2 Attack +10 Haste +2% Set: Haste +5%
Lv. 78 WAR / PLD / DRK / BST / SAM / DRG
78     Perle Brayettes Rare

[Legs] All Races
DEF: 37 DEX +5 VIT +5 Accuracy +10 Enmity +2
Haste +2% Set: Haste +5%
Lv. 78 WAR / PLD / DRK / BST / SAM / DRG
78     Perle Solerets Rare

[Feet] All Races
DEF: 21 STR +5 DEX +3 Enmity +2 Haste +2% Set: Haste +5%
Lv. 78 WAR / PLD / DRK / BST / SAM / DRG



78     Aurore Beret
[Head] All Races
DEF: 29 DEX +4 AGI +4 Haste +4%
Enhances Snapshot effect
Set: "Store TP" +8
Lv. 78 MNK / THF / BST / RNG / NIN / BLU / COR / PUP / DNC
78     Aurore Doublet

[Body] All Races
DEF: 56 STR +4 DEX +4 AGI +4 Accuracy +8
Ranged Accuracy +8 Evasion +8 Double Attack +2%
Set: "Store TP" +8
Lv. 78 MNK / THF / BST / RNG / NIN / BLU / COR / PUP / DNC
78     Aurore Gloves

[Hands] All Races
DEF: 21 STR +4 AGI +4 Haste +2%
Set: "Store TP" +8
Lv. 78 MNK / THF / BST / RNG / NIN / BLU / COR / PUP / DNC
78     Aurore Brais

[Legs] All Races
DEF: 35 Accuracy +9 Ranged Accuracy +9
Haste+3%
Set: "Store TP" +8
Lv. 78 MNK / THF / BST / RNG / NIN / BLU / COR / PUP / DNC
78     Aurore Gaiters

[Feet] All Races
DEF: 19 DEX +5 Evasion +5 Haste +2%
Set: "Store TP" +8
Lv. 78 MNK / THF / BST / RNG / NIN / BLU / COR / PUP / DNC



Teal Chapeau Rare    
[Head] All Races
DEF: 22 HP +17 INT +6 MND +6 Enmity -2
Set: Enhances "Fast Cast" effect
Lv. 78 WHM / BLM / RDM / BRD / SMN / BLU / PUP / SCH
78    
Teal Saio Rare    
[Body] All Races
DEF: 48 HP +24 MP +24 INT +6 MND +6 CHR+6
Magic Accuracy +4
"Magic Atk. Bonus" +4
Set: Enhances "Fast Cast" effect
Lv. 78 WHM / BLM / RDM / BRD / SMN / BLU / PUP / SCH
78    
Teal Cuffs Rare    
[Hands] All Races
DEF: 19 HP +12 MND +3 CHR +3 Enmity -2
Magic Accuracy +5
Set: Enhances "Fast Cast" effect
Lv. 78 WHM / BLM / RDM / BRD / SMN / BLU / PUP / SCH
78    
Teal Slops Rare    
[Legs] All Races
DEF: 32 HP +14 INT +5 CHR +5
Magic Accuracy +3
"Magic Atk. Bonus" +3
Set: Enhances "Fast Cast" effect
Lv. 78 WHM / BLM / RDM / BRD / SMN / BLU / PUP / SCH
78    
Teal Pigaches Rare    
[Feet] All Races
DEF: 16 HP +8 MND +6 CHR +6 Enmity -4
Set: Enhances "Fast Cast" effect
Lv. 78 WHM / BLM / RDM / BRD / SMN / BLU / PUP / SCH

Synthed Armor Derived from Abyssea Enemies

Created from Kulkulkan's Skin item from Kulkulkan in Abyssea Konschtat.
Grim Cuirass /+1
[Body] All Races
DEF:64 STR+15 Attack+30 Physical damage taken -5% Magic damage taken +5%
LV 80 WAR PLD DRK

Created from Briareus's Sash item from Briareus in Abyssea La Theine
Juogi /+1
[Body] All Races
DEF:52 Accuracy+15 Critical hit rate +3% "Double Attack"+4%
LV 80 MNK SAM NIN PUP

Created from two of the Regurgitated Wing items from Glavoid in Abyssea Tahrongi
Yhel Jacket /+1
[Body] All Races
DEF:38 "Magic Atk. Bonus"+9 Magic critical hit damage +4% Enmity+3 Adds "Refresh" effect
LV 80 BLM SCH

Quests
Unless you really need a party to kill something, most quests are best to do on your free time solo.  Quests reward you with cruor points as well.  Fighting enemies with a group will be a much faster way to gain cruor, but doing some quests is a good way to earn some points.  It seems like it may be possible that completing quests will tie into the zone activity or even offer NM spawn items.  So don't hurry to just write off the quests as pointless content.  Each of the three Abyssea areas also has its own fame, so repeating certain quests may be neccessary to gain access to the ones with higher requirements.  Fame levels range from 0-5 and can be viewed by speaking to an NPC at the starting camp in each zone

Time Restrictions
At first the time restrictions on entering Abyssea can be a little confusing.  But really it's just a lot like if you could use up your Assault Tags in conjunction all in the same visit.  You get your Traverser Stone key items from the NPC Joachim that act like a "tag".  Then once you enter Abyssea you will get 5 minutes of "lobby time" to hang out for a moment while you take care of any last minute chores.  Then you must speak to the Surveyor NPC and tell him how many stones you want to expend for that amount of time.  Whenever you leave the zone, your timer stops and saves your time left where it was.  You can pick up your visit time by speaking to the same NPC again any time you want.

After finishing the initial Abyssea quest "The Truth Beckons" Joachim will give you a new traverser stone for every 20 hours that passes.  Initially you can hold 3 stone key items at once.  With the right Ancient Abyssite however you can hold up to 4 (see more on that later).  Since every Traverser Stone is worth 30 minutes of visitant time status, 4 stones is 120 minutes.  120 minutes is also the maximum amount of time you can accrue to spend at any one time in a single visit.

However you can still order visit time at the NPC which expends the stone key items, then leave the area to save your time, return to Joachim and he will give you more stones should they have accrued.  And so technically it's possible to participate in Abyssea for 2 hours, then exit, re-enter and immediately order another 2 hours of time.

There is one proviso to be careful of however.  Any time you exit any Abyssea area with LESS than 5 minutes to spend in the zone you have to follow a penalty of a 1 hour re-entry time.  So just be mindful of this rule if you're planning to attend an Abyssea event.

One thing to note is that your characters Traverser Stones seem to start accumulating shortly after you trigger the addon.  I'm not sure if it's from the moment you speak to Joachim, or what.  But it's possible to return to him and find out you've already got 3 or more stones due to you.

While 4 Traverser Stones and 120 minutes are the maximum amount a player can be granted, there are still other ways to get more visitant time more often.  By obtaining rare Ancient Abyssites you can get more time credit for each stone as well as reduce the amount of time it takes for new stones to accrue.  Check out the next section to see more about these advantages.

*It's very important to note that should you ever order visitant time from the Surveyor, you must have your visitor status reapplied any time you re-enter an Abyssea area.  Failing to get visitant status and then leaving Abyssea will result in all of your ordered time to be wiped.  Your time will always be saved as long as you have visitant status before leaving Abyssea.

Edit as of 07/03/2010:
Although 120 minutes is the maximum amount of time a player can order from the Surveyor, it has been demonstrated that in through the help of dropped time extension chests players can remain in Abyssea for several hours more at a time.  These chests can give a bonus of 10 minutes each, and any player in the alliance who is alive anywhere in the zone should get their time extended.  It's not entirely clear if there's any strategy for obtaining time extension chests or if it's only luck.


Ancient Abyssites
Ancient Abyssites are key items that seem to drop randomly from mobs and NMs in Abyssea.  These are not related to the VNM tracking device, they are basically special perks for your character.  So far it appears as though these key items will remain on you indefinitely.  It's still too early to say but they appear to last for quite some time at the very least.  It would also appear as though all ancient abyssites will function in all areas, not just the area they're obtained in.  There are 8 ancient abyssites to be obtained from each area.  5 from enemies, and 3 that can be purchased from the NPC with cruor points.  The rare "Ephemeral" clionid and limule enemies that appear sometimes in Abyssea seem like they can drop Ancient Abyssites at a higher rate.

Viridian Tahrongi Abyssite Buffs
avarice: Increases the quantity of traverser stones the bearer may possess.
destiny: The bearer will experience greater fortune with golden treasure caskets in Abyssea.
lenity: Reduces exp loss in Abyssea.
guerdon: Grants temporary items upon receiving visitant status.
merit: Increases in maximum HP and MP and other beneficial effects will gain greater potency.

Scarlet LaTheine Abyssite Buffs
sojourn: Extends visitant time from traverser stones.
kismet: The bearer will experience greater fortune with blue treasure caskets in Abyssea.
lenity: Reduces exp loss in Abyssea.
perspicacity: Increases exp gained from enemies in Abyssea.
furtherance: Attribute bonuses and other beneficial effects will gain greater potency.

Azure Konschtat Abyssite Buffs
celerity: Traverser stones more rapidly replenish.
prosperity:The bearer will experience greater fortune with red treasure caskets in Abyssea.
lenity: Reduces exp loss in Abyssea.
reaper: Increases cruor gained from enemies in Abyssea.
merit: Increases in maximum HP and MP and other beneficial effects will gain greater potency.

Ivory Abyssite Purchased From NPCs
Tahrongi NPC
sojourn: Extends visitant time from traverser stones.
confluence: Reduces the cruor cost of using veridical confluxes in Abyssea.
expertise: Allows more frequent use of Martellos.
Other (it's unclear where these other two might be obtained)-
fortune: The bearer will come upon treasure caskets more frequently in Abyssea.
acumen: Bestows the bearer with knowledge of Abyssean treasure casket locks.

Notorious Monsters
Abyssea areas are populated with different tiers of NMs.  Some of them are already pre-spawned and just hanging around.  While most of them require a spawn trigger to appear.  In most cases you can easily track down the type of enemy you need to spawn an NM of the same type.  Then just farm a few of them until they drop the NM trigger you require.  The first tier of NM triggers can be bought from the AH even.  At the second tier however NM triggers can become a rarer drop and most are key items.

This is important because obviously you can't trade key items, meaning the same person has to have all of the required items to spawn the NM.  Sometimes this takes several different things.  If the key item drops when an NM is defeated it seems to be given to the person who first spawned the NM or aquired enmity upon it.  Be careful of this so that you don't wind up with someone pulling a certain NM who you don't want to get the key item (in the case of NMs that are pre-spawned).

All NMs with items required for magian trials can drop up to 2 at once (drop rate seems fair but not 100% similar to T3 VNM)  These are used in upgrading the Empyreon Weapons featured on the Visions of Abyssea website.

Each Abyssea NM will have special weaknesses that can be exploited in order to hinder their movement or limit their special actions.  According to the NPCs these weaknesses' fall into the broad categories of magic attacks, physical-based weapon skills, and magic-based weapon skills.  It's not clear yet whether NM weaknesses are always different or whether each NM has its own unique weakness.  But try different things out and you might find a way to mollify an extremely tough enemy.  You will know that you've found a weakness if you see a "!!" double exlamatioon graphic appear above the NMs head.  What's more is that finding these weaknesses to defeate enemies rewards you with Atma buffs.  NM weaknesses' are still laregly unknown and untouched so please do experiment.  Read on for more on Atmas.

Atma
Atmas are unique key item buffs rewarded for exploiting an enemies weakness and successfully defeating it.  Once you've obtained one you also need a Lunar Abyssite key item to have it activated.  You're awarded with a Lunar Abyssite after finishing the main Abyssea quests To Paste a Peiste, A Goldstruck Gigas, or Megadrile Menace.  With an Atma and Lunar Abyssite speak to the Atma Infusionist NPC to have it activated.  I'm not sure but I presume you can only have one Atma activated at a time.  There are a total of 30 Atmas in all, 10 from each zone.  Collect 'em all!

LaTheine-
Atma of the Lion
"Triple Attack"+: Minor
Physical damage taken -: Minor
Lightning elemental attack +: Major≻

Atma of the Stout Arm
STR+: Major
Attack+: Superior
Ranged Attack+: Major≺≻

Atma of the Twin Claw
Defense+: Major
"Magic Def. Bonus"+: Minor
Enhances "Resist Charm" effect: Minor≺≻

Atma of Allure
MP+: Major
MND+: Major
Enmity-: Major≺≻

Atma of Eternity
Enmity critical hit rate -: Minor
Enhances "Resist Slow" effect: Major
Enhances "Resist Curse" effect: Major≺≻

Atma of the Heavens
Magic Accuracy+: Major
Physical damage taken -: Minor
Enhances "Resist Paralysis" Effect: Major≺≻

Atma of the Baying Moon
Obtained from Lugarhoo
Attack+: Major
"Magic Atk. Bonus": Major≺≻

Atma of the Ebon Hoof
HP+: Major
Enh. "Resist Sleep" effect: Superior≺≻

Atma of Tremors
Damage taken -: Minor
Enhances "Resist Silence" effect: Major≺≻

Atma of the Savage Tiger
Obtained from Megantereon
AGI+: Major
"Double Attack"+: Minor≺≻

Tahrongi / Konschtat-

Atma of the Voracious Violet
STR+: Superior
"Double Attack"+: Minor
"Regain": Minor≺≻

Atma of Cloak and Dagger
Accuracy+: Major
Evasion+: Major
"Death" resistance: Minor≺≻

Atma of the Stormbird
Wind elemental accuracy +: Major
Lightning elemental accuracy +: Major
"Refresh": Minor≺≻

Atma of the Noxious Fang
"Subtle Blow"+: Major
Water damage attack +: Major
Enhances "Resist Poison" effect: Major≺≻

Atma of Vicissitude
Defense+: Major
"Magic Def. Bonus": Minor
"Regen": Major≺≻

Atma of the Beyond
"Magic Atk. Bonus": Major
Ice elemental attack +: Major
Light elemental attack +: Major≺≻

Atma of Stormbreath
VIT+: Major
Breath damage taken -: Major≺≻

Atma of Gales
Wind elemental attack +: Major
Wind elemental accuracy +: Major≺≻

Atma of Thrashing Tendrils
CHR+: Major
Critical hit rate +: Minor≺≻

Atma of the Drifter
Ranged Attack+: Major
Ranged Accuracy+: Major≺≻

Tahrongi / Konschtat-

Atma of the Stronghold
Attack+: Major
Defense+: Major
"Regen": Major≺≻

Atma of the Harvester
STR+: Major
"Double Attack"+: Minor
Enhances "Resist Sleep" effect: Major≺≻

Atma of Dunes
Enh. "Drain" and "Aspir" effects: Major
"Store TP"+: Minor
Enhances "Resist Slow" effect: Major≺≻

Atma of the Cosmos
Darkness elemental attack +: Major
Enhances "Resist Amnesia" effect: Major
Enhances "Resist Silence" effect: Major≺≻

Atma of the Siren Shadow
Attack+: Major
Evasion+: Major
Enh. "Resist Paralysis" effect: Major≺≻

Atma of the Impaler
"Double Attack"+: Minor
Enhances "Resist Bind" effect: Major
Enhances "Resist Blind" effect: Major≺≻

Atma of the Adamantine
VIT+: Minor
Defense+: Major≺≻

Atma of Calamity
Enhances "Resist Slow" effect: Major
Enhances "Resist Blind" effect: Major≺≻

Atma of the Claw
Earth elemental attack +: Major
Earth elemental accuracy +: Major≺≻

Atma of Baleful Bones
STR+: Minor
Darkness elemental accuracy +: Major≺≻

VNM 

Yes that's right.  As if there weren't already enough to do, Abyssea areas also contain their own VNM paths in each zone.  Instead of using a clear abyssite however you need a "clear demilune abyssite" key item from the NPC for 300 cruor.  Little is known about these VNM paths so far, except that they are Empty type mobs (the T1 are very easy).  La Theine clear abyssite upgrades to an scarlet abyssite when it reaches T3, while Konschtat becomes azure, and Tahrongi change to viridian.

Chests
In Abyssea you'll find lots of chests containing different items and bonuses.  Just like the golden caskets randomly dropped from enemies in Conquest areas most of them are locked and requiring solving a mini puzzle to open.  This time however there's two new types of locked puzzles to overcome.  Just beware that while chests dropped from enemies will always be safe, some of the Sturdy Pyxis type chests pre-spawned can turn out to be very strong Mimic enemies.  They are very annoying and not worth wasting your time on.  It's possible that the pet deaggro trick can work as a cautious way of checking for Mimics.

Failing to open a chest correctly can rarely result in a small time penalty (1 minute it seems).  On the other hand some can even add time onto your visitant status (10 minutes has been seen before).  After examining a chest you can choose the "peak inside" option to get an idea of the reward, then proceed with unlocking it or ignore it.  The effort can definately be worthwhile as chests can grant hundreds and sometimes ~1000 points of experience or cruor as well as augmented items and more.  If you're going to be busy fighting furiously in a large group it's a good idea to assign one person as the chest opener (usually the one amongst you who didn't flunk math, that guy).

Occasionally you might find a Forbidden Key item randomly dropped from enemies.  This can be used to open one of the Sturdy Pyxis chests hassle free.  You will also be rewarded with temporary items from certain chests and enemies.  Keep in mind temp items you gain will be retained if you leave and return again in another visit someday.  The color of each chest designates its contents.  Gold chests contain items, blue chests grant experience/limit points or temp items, and red chests give cruor or different light auras.

Light Auras
After defeating an enemy you'll notice a message about emitting a "light" of some kind.  It's still fairly mysterious and unknown what exactly these "light" messages are for.  There are 7 possible colors of light: azure, ruby, amber, pearlescent, golden, silvery, and ebon.  Some have observed that the most common types depend upon the killing blow dealt to the enemy.  Using a weaponskill gives the ruby light, a melee attack pearlescent, and a magic attack azure.  Some types like golden may only come from locked chests.

These lights also vary in strength ranging from feeble, faint, mild, strong, and intense for the azure, ruby, and amber types.  While pearlescent, golden, silvery, and ebon only rate from faint, and mild, to strong.

Conflux Warps and Martellos
Strategically placed around Abyssea areas you'll find 8 Conflux warps and 8 Martellos.  The warps require a one-time activation with enough cruor points which from that point forward allows you to travel quickly from one to another.  They cost a little cruor to travel around from, but they can save so much time walking around such a huge area they're vital to save you visiting time.

Martellos can be used to instantly heal you HP, MP, or status effects.  And so it can be a good idea to fight a difficult NM near one for a quick refill in an emergency.  You can only use the a Martello once every 30 minutes so plan accordingly.  The Martellos perks are expended at the cost of fuel, and once it runs out it will no longer offer its service.  This hasn't ever been a problem in my experience, and there are different quests that refuel a Martello as well as upgrade its capacities.


That's about everything I can manage to point out for now.  Keep in mind some of this info is still new and mysterious so it could wind up incorrect or irrelevant eventually.  I hope all of this can assist you and help overcome the challenges you find in Abyssea.  I might post some zone specific info next, but in some regards that info is still being figured out and it would be incomplete.  If you find anything new or interesting be sure to post it over at the BG forums Abyssea thread.  Have fun.