you'll find plenty of reasons to explore the maps as you progress. fates and leves will take you all over if you're grinding exp (which you shouldn't do until you're leveling a second class; the main story gives you enough exp for your first class), but you'll also be running everywhere if you start playing a gathering class.
you'll start unlocking dungeons shortly after you start riding airships. keep in mind this game has tons of content, that unlocks as you play the game, including stuff you can't access until you beat the 2.0 story. there's crafting and gathering classes, daily quests and roulettes, multiplayer, the gold saucer, house building, chocobo racing, triple triad, GLAMOUR, etc etc etc. this is a huge game and it'll be a long long time before you exhaust all the content.
as for the story, yes it will get more interesting soon, although some of the level 20 story quests have the worst fetch questing bullshit ever. like pretty much everything else in the game, it gets better and better the further you go.
and as for it not feeling like an ff game, just you wait
there'll be so many fun references, homages, and cameos that you will flip out if you know your series history.
