I re-started a little over a week ago now, and I'm pushing 50 on my Dragoon. Currently 47. Here's my advice.
Ignore all side quests, with the exception of the ones that are colored blue. The blue side quests unlock various things in game, like the ability to dye your gear, certain dungeons, palace of the dead, etc.
Keep on your Main Story Quest and Class Quests (class quests are every 5 levels). Keep up with your class hunting log and do Fates that are on the way when you are going about the MSQ.
I went from 1-35ish just on MSQ and few fates, class quests and the lancer hunting log.
Also, do your daily roulette (aka dungeon) for the bonus xp. I also pop a food buff for the extra XP.
Post 35, I had to supplement Palace of the Dead with MSQ and Class Quests.