Most of your confusion lies around the (completely unprecidented) abyssea system. Atmas, lights, abyssites, new exp chain-system etc. You can get everything off of wiki pretty much as it is all detailed Here
*Atmas are key items that give you 2-3 large buffs each and only work in abyssea. Once you get one (by killing X number of zone bosses or buying with the 'abyssea points' Cruor) then you can pay a nominal cruor fee to 'equip' it. You can get up to 3 once you have met certain requirements. These are huge boosts like 10-tic refresh, +50 to a stat, +50 to atk, crit damage/rate+30%, MAB, HP+30%, Element X resist+100, +double/triple atk, TP Regain, pretty much anything you can think of is available. These HUGELY impact your performance and can be tailored to whatever job/roll you want to fill once you have access to equip 3 and have a variety attained.
Once you can equip atmas, you can obtain additional atmas from NMs inside abyssea, quests, Titles outside abyssea etc.
*Abyssites are also key item buffs that only work in abyssea, but they are 'always on'. Things like increased exp in abyssea, more cruor, more HP and such. You also get these from NMs/Quests/Stuff in general in abyssea.
*The !! monster weakness system. Performing certain actions (as rng it will be certain WS at certain times) to trigger a monsters weakness. Learn what WS you can use that might trigger it and you can be far more useful to groups.
*The light/chest system is important to understand, and is detailed on the wiki. Make sure you figure out how different types of damage make certain lights and what those lights do and how it relates to the chests you get.
Its a LOT of info all at once, but if i had to prioritize to do it all over again:
#1 seek out all 3 abyssites of celerity (can look up abyssites on wiki too). These reduce the recharge time of your entry stones so you can enter abyssea a LOT more often. With all 3 you get 3 stones a day instead of 1. Triple the time in abyssea go!
#2. Get as many Zone Boss kills as possible. There is 1 special NM in each zone that gives you credit for the 'major' quest/mission line in each of the 9 total zones. You get your 1st Lunar Abyssite from your 1st boss (lets you equip 1 atma). The 7th also gives you a second (can equip 2 atmas. 3rd you buy with cruor.). They give other rewards with each successive kill. It doesnt matter what order you kill them in.
#3. Abyssites of Sojourn: Each extends your time by 3 minutes per stone. so in stead of 30 min/stone, you can get up to 48min/stone if you get all 6. Some are easier to get than others, but pick them up when you can.
If you do those 3 things, you will be fairly well setup. You will be able to spend far more time in there to farm up lewt and with 2-3 atma slots (lunar abyssites), you will be able to buff yourself up immensly. After that, you can just farm up additional atmas, abyssites and gear (new AF gear and random other gear) far more effectively.
Just familiarize yourself with the linked wiki page (general abyssea info. Chests, monster weaknesses, lights etc), and after you got that down, start looking up what atmas and abyssites do and track down the good ones after you have the 1st 3 things moving along.