The first dungeon is level 25 (The Thousand Maws of Toto-Rak), you'll need at least one other person to enter (up to a maximum party of four people total), and you must be a recruit in a Grand Company in order to access it (I think?).

I'm not sure why you equate FF with dungeons though. There's only... 4 of them in the game right now, and aside from Toto-Rak, they're all level 45+.