Welcome to Eorzea!
There is definetely a lot more to do than just questing - the way the game is designed though is it specifically meant to unlock new content and gameplay gradually as you go along in the main storyline. This includes dungeons - the first instanced dungeon is unlocked during the level 15 main story quest so you will get there soon enough.
Other content you can do right from the start at your point in the game is FATEs, these are basically open battles or scenarios out in the field that have a basic objective to complete before time runs out, this can be as simple as "kill this big monster" or "pick up items off the ground and give them to the NPC". FATEs reward EXP, gil and, once you join one, Grand Company seals. You can freely join and leave FATEs as much as you like, so if you lose interest there is no penalty. FATEs are identified by distinctive pink and blue icons on the map.
Really though, you are very early in the game so you have barely dipped your feet in the pool that is this game yet. There is a
lot to do, but the game does a good job at not overloading the player with it all at once, as it's a journey you proceed on. Have patience, and you'll soon find new opportunities unlock you weren't expecting.
