As most people mentioned the daily Low Level Duty Roulette reward is the single biggest source of experience (it's like something around 45% of your experience bar for each level).

Aside from that, someone else did mention the challenge log but I just wanted to emphasize that the challenge log is probably the second biggest source of experience. I just took my Warrior from level 41-50 in somewhere around the neighborhood of 5 days by focusing on the challenge log. Do your dungeons, give commendations, do guildhests, do levequeusts, and do FATES. Each one of those things individually gives decent XP but when you complete it's challenge log entry it really boosts you up - and it forces you to change up what you're doing so you shouldn't get too burned out on a single activity (or if you do you can switch since there's a lot to do). If you don't have your challenge log unlocked yet I highly recommend you go make that a priority.

The challenge log directly points you to activities that earn you experience and then rewards you even more for doing those activities. Focus on that (it resets once a week) and you'll be gaining levels very fast.