I will second that the MSQ is by far the most important thing to do. The MSQ unlocks(either directly or via gating) most of the content in the world. There are some parts of the world you cant even get to and explore until the MSQ sends you there. So thats definitely priority one. Unlock quests are an ok priority two. If you want an example of one, your character lists you as an archer. There is a level 15 quest in Gridania(you have to advance the story a bit past 15 for it to show up i think) with the blue icon. If you do that quest, you unlock the sightseeing log(a sort of mini-game where you have to find specific spots in the world and interact with them via emote). A similar level 15 quest in Ul Dah unlocks access to the Manderville Gold Saucer(a casino full of mini-games and non-combat rewards). Youll find a lot of them over time. Finally, a third thing to prioritize is your class/job quests. Some of your most important combat abilities are unlocked by doing class/job quests, so you definitely dont want to skip those either.

Welcome to the game, and happy hunting!