Basically go to each nation and look for the related VW NPCs. "Sign up" or whatever the specific option is and you should be given a colored stratum abyssite (Jade, Indigo, and Crimson for the cities). At this point, you should also start accumulating stones, which you get replenished over time at the NPCs or by buying/finding Voiddust. With these, you then go out into the field to pop your mobs.
People tend to refer to the mobs in tiers, and tier 1 involves the mobs in the zones right outside their respective city, past and present. Defeat both for the respective region and can you get your abyssite upgraded at the NPCs when having it inspected. You're now capable of popping T2 mobs (Gusgen, Ordelles, or Shakhrami). You only need to kill one of these to make your abyssite eligible for upgrade to T3. Pashhow, Meriphitaud, and Jugner house the next mobs, past and present. Beat both for a section and it can become T4. These then let you pop the mobs in Horo Ruins, Wadi, and KRT. And just to be clear, having Windy's T4 will not let you pop Bastok's T2 or beyond. You need that specific path's abyssite to pop the respective mobs. Once you have it, its yours for good. A T4 can pop T1 mobs fine.
Once you get all three of those maxed and the T4 mobs killed, head to Pashhow in the past for a CS at the conflux near the telepoint. This flags you for a bit of legwork to access the Jeuno and Zilart abyssites. First begins by talking to a VW NPC in one of the cities. Other should begin by talking to a guy in Norg near the docks. Just be sure to talk to people in Kazham and Rabao for the Zilart path to unlock some teles and I think finish flagging you for those fights.
Anyway, I haven't gone into detail for the fights since that's really something the people you're doing them with should explain. If you're doing a fight for the first time, you'll probably want to feed the void 3 red cells to boost your red light. This will help you get KIs and Atmacites which can make future fights easier. Blue cells will improve the possible number of items you can get, which might be nice to do on some mobs that still have pricey drops. Pick a wiki and research. Green/Yellow handles Cruor and EXP. Up if you want.
The fights themselves pretty much become hunts for weaknesses. When you first get your T1, you can buy and abyssite that'll give you hints on spells and WS as you use them. You'll get messages pointing out (high) vulnerability to a magic element, JAs, or weapon type. Landing one of these will improve a particular color on the spectrum, with high procs yielding more points, freezing the mob for a short bit, and replenishing temp items if you have those KIs. In effect, you'll want to have all your spells and possible WS, even for sub-weapons you might not normally use.
By the time you're T4, hopefully you should have a hang of the system. All that will vary then is how the mob fights.