I'm going assume you mean unlocking a new class (Not DRK/AST/MCH)
Your first step is the Hunting Log. Completing these along with clearing any FATEs you come across should push you up to level 15.
If you find yourself short a bit, do the Guildhests and Levequests. Guildhests especially because each class gets a large bonus to EXP upon first completion.
Post level 15, you have two choices. The first is the (IMO Sensible) running of Dungeons. Dungeons give quite a lot of EXP once you factor in the 100% Armory Bonus for having a class at level cap.
The second choice is FATEs. Endless FATEs.
The better choice is to do a mixture of FATEs and Hunting Logs while queuing for Dungeons.
Don't forget to run your Daily Roulette for the large chunk of EXP there.

And finally, don't try to rush to cap in a day. Just take your time and cruise. The 'grind' doesn't feel quite so bad when you're relaxing.