There are other options besides doing quests to level your character. You can reach level 90 without doing more than a few quests. The only ones that are required are the ones to enter new content such as the Mists of Pandaria beginning quests. There are a lot of reasons you might want to avoid questing. You might have a max level character willing to help you, you might be sick of doing the same quests over and over, or you might be looking for a way to make some extra gold.
If you’re level 20, I recommend heading to the south bank of the river separating Elwynn with Duskwood. Stay there until you’re level 21 and then check out the maps below:
Duskwood (Level 21-27)
There are three really good areas you can farm in Duskwood. There is actually a fourth, which we used in the level 10-20 grinding guide; but there are three you can use in the 20s. The first is to the east of Twilight Grove “crater.” You’ll find worgen, spiders and wolves. Once you reach level 23 head south to The Rotting Orchard, where you’ll find the same types of mobs (worgen, spiders and wolves). Finally, at level 25 you can head to the Raven Hill cemetery where you’ll find more spiders and some undead lurking around the graves.
Northern Stranglethorn Vale (27-30)
Duskwood is only good until about level 27, once you ding you can head south to Northern Stranglethorn Vale. Here you can grind to level 30 in and south of the Ruins of Zul’kunda where you will find trolls, panthers and raptors.
Tip: Have you ever wondered how players have so many level 85s or how they can level an 85 so quickly? I use and strongy recommend Zygor's in-game leveling addon which will enable you to level in 5 days of played time. You can find a free version of it here.