1-15:
Run your storyline quests and participate in occasional FATE's along the way. It's kind of boring, but you can get through it pretty quickly.

Regarding FATES: don't be afraid to pull out if the FATE is scaled all out of wack. There's no reason to spend 7-8 minutes on a FATE that is going to give you 2k exp... when you can spend 45 seconds running to the next town and pick up 5-6 quests that you can complete in 7 minutes that are going to give you 10-20k exp and gil/items.


16-50:
1) Every day, run your duty roulette. First thing you do, run those duty roulette's. That's your biggest bang for your buck.
2) Quests (storyline and sidequests), focusing on your hunting log in each area along the way
3) Dungeons
4) FATE's (I like to knock off easy FATE's en route to/from storyline quest objectives)


Regarding questing... pick up EVERY quest from an area, and knock them all off before returning back to the settlement. Many times, you'll get additional quests that will send you right back out into the world. It's better to knock 3-4 of these off at a time, as opposed to running back/forth over and over turning in individual quests.