Having so many alts, I have a little experience with all the classes. Except warriors. I've never gotten a warrior past level 10. However, I feel confident in stating that priests are probably the toughest class to level. I love playing my priest - standing at the back of the group, trying to keep everyone alive, is a fun (depending on the group, of course) challenge, and the priest certainly has the bells and whistles needed for healing. Levelling one, though, makes me sad (my paladin wasn't the only character I nearly deleted at 20 and used as a bank for awhile).
I talked about how specs are important for levelling speed in a previous post. I don't know if you'll find a better example of that than the priest class. I suspect most people start levelling a priest because they want to play a healer, so there's some resistance to speccing any other way than Holy. I can confirm that it's a long, hard slog if you're going to level that way, though. Stuff takes forever to die, and so it's easy to run out of mana between smiting your mob and keeping yourself healed. So unless you're levelling with a group or by pugging instances, I think you should steer clear of a holy build.
Shadow has been the talent tree traditionally used by levelling priests. Mobs die quickly, there's a fair bit of mana regen, and mind flay is the most mana-efficient damage spell you'll ever wield as a priest (especially in shadow form). Elysaa, my priest, levelled most of the way from 20 to 70 as shadow. It worked pretty well, especially in the high 40s, but after that, survivability started to suffer somewhat. I turned back to shadow once I decided my dual-boxing experiment was over (I had been levelling her as holy alongside my affliction warlock), and was very happy with the damage. I did find that I was dying a lot, though. If I picked up a second mob, I'd start running into mana problems as I fought them and kept myself healed up. Keeping myself healed would take me out of shadow form, losing me a lot of mitigation and damage unless I could expend the mana to pop back into shadow form again. If a third mob showed up, I was probably going to die.
Other priests in our guild suggested I go discipline. I had looked at discipline, and have always liked the tree (I started playing months after Burning Crusade, so I don't know what it was like earlier). Our earliest discipline advocate had switched over last summer because it was the best PvP tree, and he spent most of his time in battlegrounds. A later addition, who only started playing after Wrath launched, had levelled to 80 mostly as discipline, and found it to be very good. After playing with shadow for a day, I respecced to discipline.
Here's the discipline priest levelling spec I'm using right now. Unlike shadow, a build like this is good for levelling and healing instances. There are a few points that could probably be moved around to suit another playstyle, but I think this is an all-round good levelling/healing spec. Now, don't get the idea that I'm jumping on the discipline bandwagon. I still prefer holy for healing instances, and will go back to it at 80. In the meantime, discipline combines damage and survivability in one tree like no other.