You dont have to do quests at all if youre getting sick of them.

Why would you want to do quests if you're tired of them? If you want to go roam and fight, go roam and fight. The only restriction to roaming is if you started in Limsa, which according to your data, you started in Ul'dah, as a Thaumaturge which later becomes black mage.

Basically at level 15 you unlock airships so you can travel between all starter cities.

Best advice is do not do every single quest you see. Especially if youre getting sick of them and youre not even level 30.

Go out and have fun, if youre doing something that isnt fun, change it up. There is an insane amount of content, duties, ways to level, FATEs, challenge log (level 15) the list goes on. Dont bottle neck yourself and get burned out before you even get started.

View Raldo's post, those are the only important quests. the rest are all optional.