Specifics for 11-20
I was level 20 well before the quest "Pulling Back the Veil" which
I farmed relic weapon pieces in the Oblivion Woods for my Mystic, as well as farming Stonebeaks for the Vanguard Quest (repeated this quest until I maxed my daily Requests, 8 I think).
Doing this nets you around 2,500-3,000 exp for each completion, and depending on class, can be done fairly quickly (literally less than a minute).Some helpful tips for leveling:
* Quests, especially main quests, the most obvious way. You should do main quests until level 20, then dungeons.
* Dungeons. Next to main quests. Lots of enemies and bosses, you can get a large amount of exp out of these (first dungeon is available at level 20, combine with the Rested Bonus (below) for even more exp potential).
* Vanguard Requests. These offer a nice exp boost and other items upon completion, and are available early in the game. Check often, as you get more rewarding requests with higher Item Levels
* Rest Bonus. If you go offline in a town, you start to accumulate rested bonus. Killing monster while "Rested" gives DOUBLE exp, quests do not grant double exp.