You should have unlocked the airship at Lv.15. If you haven't, finish your story quest asap. The story quests are indicated by a special flame quest sign, as opposed to the circular quest sign for non-story.

Anyway, leveling another class is just like starting from Lv.1 again. You do gain one cross-class slot from the outset, and a 50% xp bonus from FATEs and killing monsters, but that's it. There's no bonus to quest xp. You can switch to your main class at any time by equipping your weapon (be sure not to discard or sell it). The biggest penalty to leveling an alt is that you will not have access to previous quests (as quests outside of the beastman dailies and leves are not repeatable).