First off, I believe levelling quickly in the "power-levelling" sense, and levelling well, are two different things. Getting a level 60 friend to run you through all the lower dungeons for loot won't help you learn how to play your class in groups; spending five levels grinding in one spot doesn't equip you to deal with a wide variety of situations. The journey through each level should, in my opinion, be savoured and learnt from -- after all, when you end up at level 60, knowing as much as possible about your class is a great start.
There are arguments for and against levelling through questing, levelling through grouping and levelling through grinding. I find that questing solo gives the most XP; after all, you get the XP for killing monsters plus a huge chunk when you turn the quest in. However, it helps to stack quests; that is, work on multiple quests in the same area at the same time. Running backwards and forwards to quest areas and NPCs takes up a tremendous amount of time, so if you can minimise it then you will find you level faster.
Jame's Alliance levelling guide, which starts at level 30, is a masterpiece of quest stacking and maximum efficiency. You can follow the guide from start to finish, or read it for tips on which quests may complement the ones you are currently on.
Grouping up for instances or elite quests is a great way to get them done early, maximising XP and the use you will get out of any items dropped. Instancing also makes a nice change from soloing, and helps you prepare for the endgame; it's not as efficient XP/hour wise as questing or grinding, but it can break up some monotonous levels.
Plain old grinding -- setting yourself down in a spot and killing anything that wanders past -- is what some people swear by for maximum XP/hour. By not running to and fro, you can kill extremely efficiently and get a lot of XP. You can also choose mobs that are easy for you to kill with minimum downtime; a couple of levels lower than you works well, and humanoid mobs tend to drop cloth and money which will help you get rich too. I personally don't like the monotony of grinding, but when I'm close to levelling I have been known to grind the last few thousand XP rather than spend a while flying to hand in a quest.
A couple of things that have helped my current character level quickly, and the time fly:
- Other players. By keeping track of a number of similarly-levelled players, I could group up for a lot of difficult tasks, making the XP pour in.
- Rested XP. I try to keep at least some rested on all my characters, and rotate amongst my alts to use it up; even a little bit helps.
- Equipment. By researching equipment on sites like wow-loot.com, I was able to prepare and gear myself well early on, meaning a more powerful character and thus faster levelling.