There are lots of core aspects I like about this game but I'll focus on the activities I enjoy to do every patch.
As a collector I'm always busy. I'll grind away during seasonal events if the reward is worth the effort. I have also done hunts, Diadem, Gold Saucer, Gordias as a tank, ARR primals, friend recruiting, gardening, Aquapolis, PotD and beast tribe quests for mounts and minions. One of my long term goals used to be getting gil for a house, then another house and now furniture. I'm at the stage where making gil is starting to pay off because I can buy most minions and bardings on the MB. (When I say "making gil" I mean quick synth during breaks.) I also collect glamours and currently one of my retainers is tasked to carry all that crap. It never ends... The level 60 relic is probably the hardest glamour to get at the moment. The worst part is that we don't know the final forms yet and I can't spend any relic items before I know what it will look like in the end.On top of the aforementioned it's always fun to get a new TTT card or Orchestrion roll.
This is somewhat related but still a bit different to collecting. I'm a completionist. I try to get as many achievements, log entries, quests, reputations, maxed classes, chocobo ranks and Aetherytes. Everything that is permanent to a character is worth getting to me. Whenever I get tired of collecting things I level crafting class, a battle job or my chocobo, do some sightseeing or continue an ARR relic quest (I need that Z).
All of those activities supplement the daily and weekly battle activities I do on my main jobs. I have no real reason to gear up at the moment so I do battle content out of the sheer enjoyment of playing a certain job.
Sometimes when I'm up for a change of routine I make alts or play on one of my alts on EU and NA servers to relive the story. It's strange to see the differences in server cultures and data center cultures. An example: on the Aether data center it's very common that dps go full ham on everything that moves in low level dungeons regardless of whether they exceed the tank's aggro, whereas on Chaos people are more considerate towards the tank's role. Neither is wrong, but these two communities just place different value on dps/speed and traditional MMO party etiquette. As for server culture, I like how on the NA server people people talk all the time. Chit chat everywhere.