With characters both on Gilgamesh and Balmung for RP, it boils down to what your comfortable with.

As people have said, on Balmung RP is EVERYWHERE. You can literally go to some of the Adventurers guilds (most popular is the Quicksand in Ul'Dah) and just randomly talk up people in character. If you're comfortable with that, that is. You can also get some people that are totally troll RPers, so I've found as a serious RPer that has trouble with random encounters, it's really daunting and hard to get involved, and it's not even easy getting into an LS or FC dedicated to RP that doesn't start up and fizzle away (I've had 2 FCs I've joined since I got on right before heavensward just die.) If you like the large random and just get in thing, this place is AWESOME.

On Gilgamesh... well everything is focused through the FC and LS communities. There are several dedicated to RP and if you know where to look, you can find a lot going on. That said, I find that they have a lot of cliques and while it's easier to jump in as all RP ends up as a mini-event, it's hard to get anything really character focused going because people like their big events like bar nights and parties and The Clinic (which is an IC place for people to go and get injuries treated). I'm still working on getting a real foothold, but because the community is more centralized, it's at least easier to get a conversation on the topic started, even if I'm not getting results.

There are also small communities on other servers. The one on Lamia is FC based and really laid back and casual, so if you want something smaller with a feel of casual interactions and conversation it's a good spot. There's also one on Brynhildr (it's been a while since I've been there so I don't know it's current status) but they love their great big story arcs and it's really interesting. I ended up not being able to follow all the huge things they kept pulling though and that was one of the reasons I left. There was once a community on Siren, but I think that may have up and left for Balmung.