The best way to level is always doing your Main Story Quests (the ones that look like they have a little fire around them) and the dungeons they unlock At these lower levels especially, really focus on your story. As Hasrat said, make sure to do the Hall of the Novice as well. I was new to MMOs when I first started, so I kind of thought of the Hall of the Novice as like DPS lessons and found it really helpful, plus you get pretty decent gear for when you unlock your first dungeon (level 15 Main Story).

That being said, I would also suggest finding a new player-friendly Free Company. I'm not familiar with the Moogle server at all, but I'm sure you'll find one if you look around Generally people are pretty nice and willing to help new players learn the game and answer any questions you may have. And if it's your first time in a dungeon or anything, the other players can see that you're new, and most are willing to help if you ask for it. Good luck!