How can I quickly level up in World of Warcraft?

There are a few things you can do to accelerate levelling:

Dungeons Use the Looking for Group tool to queue for dungeons. There are often quests within dungeons with good experience rewards (the first time at least). Dungeons also have high level mobs which will have a good rate of experience on them. Loot from dungeons will also make your character more powerful, making questing easier/faster.

Quest wisely Move out of areas once you out level them - don't feel compelled to stay and finish the story lines if your priority is levelling.

PVP Some people get a good rate of experience/hour by winning in battlegrounds.

Have a max-level character Your colleagues will have advantages over you if they have access to Heirlooms. As well has having excellent stats some heirlooms have bonus experience for both killing mobs and turning in quests.

Rest bonus Make sure you are in an Inn or a capital city when you log out to accrue rest bonus.

Gathering professions Nowadays you gain experience when you gather herbs and mine ore. You can then sell the herbs/ore on the Auction House and buy better items to level faster.

Guild Perks Probably the easiest action for a new player is to join a guild and take advantage of the levelling perks, especially Fast Track (lvl 6 guild). Other useful perks for levelling are Mount Up (lvl 3), hasty hearth (lvl 8) and working overtime (lvl 14). wth thanks to @Jereone for the the reminder

Questing is the fastest way to level without Recruit-a-Friend. Dungeons work well too, especially if you haven't done the quests for the dungeon that you get in. Battlegrounds take way too long to be done exclusively in my opinion.

With Recruit-a-Friend, dungeons are a great way to level, and you can power level two characters very quickly if you have a max level character on another account (You, friend, etc). The max level character can just kill everything quickly while the two low level Recruit-a-Friend accounts follow behind.

Are you questing in zones that are your current level? Meaning that the quests are not green or grey in your quest log. It took me less that 74 hours to level a priest to 85 without heirlooms just recently. I did a mix of quests, dungeons, and battlegrounds on the way.

Another thing to consider is using a quest guide - the WoWPro guides or possibly (QuestHelper)[http://www.curse.com/addons/wow/quest-helper]. Both of these are available as add-ons and will help you to do quests in the most optimal fashion so you don't spend a ton of time running back and forth.

Get the fastest mount you can. You should be able to afford a 100% mount. The less time you spend traveling, the more time you're questing and getting XP. If you don't use a leveling guide, bind your hearthstone to wherever your questing hub is located. Use your hearthstone to get back to turn in quests once you've collected / killed everything for the current group of quests.

One other tip: Get the biggest bags you can afford. You should be running at least 16 slot bags in every slot. Before you head out questing, dump everything out of your bags that you can. The more bag space you have, the less often you have to sell or deal with inventory management. Also, get a junk selling add-on.

Disclaimer: I've used the Wow-Pro guides for a while. They're pretty good (both the ones on the site and the add-on version). Never used QH, but I've heard lots of good things about it.