Firstly be aware that I did most of my leveling pre-3.0.2 patch. Some of these things may have changed slightly since then. Mana regen is the most likely change as this seems to have been boosted significantly in 3.0.2. Now on to the tips.
1. Put your first talent points into Spirit Tap. Regardless if you plan to level as a Shadow Priest or a Holy Smiter (also known as a LOLSmiter, because they used to be relatively weak), always take this talent first. It will save you hours and hours of waiting for mana regen, especially at low levels where you don’t have much.
Most players will find it easier to level, at least as far as level 50 or so, using a Shadow spec. In this range Shadow has better damage and you can heal instances fairly easily. Post about level 50 you choice of spec depends more on your play style. Many players stay shadow all the way to the level cap. Personally I respeced into a discipline/holy build. This used a couple of key holy talents like Holy Specialisation and Divine Fury to increase damage output from Smite and Holy Fire while speccing into Reflective Shield for a nice bit of passive damage every single pull. This also continued to allow me to be the healer in instances quite easily. It is of course personal choice and by the time you get to level 50+ you’ll be better informed anyway.
2. Get a good wand. Especially pre-30 a good wand will make levelling much much easier. Early on you will be casting Smite (or Holy Fire@20), followed by Shadow Word: Pain. Then auto attack with your wand. At some levels, with a decent wand, you will be doing far more damage with your wand than with your spells. Also the wand auto attack can’t be interrupted, unlike spells, which is important until you level a bit and can afford to cast Power Word: Shield at the start of each pull.
You can usually buy a good wand on the Auction House quite cheaply. Otherwise buy a Simple Wood (enchanting suppliers sell them) and a Lesser Magic Essence and ask for an enchanter to make you one. Some enchanters will supply the essence (which they have plenty of) and will make it if you supply the wood (which they usually don’t carry around with them).
3. Quest a lot. Well duh?!? To get the fastest levelling quest like a demon. Yes you can level by grinding and doing instances but questing is by far the fastest. Two useful tools here are the zones by level wiki page, for knowing what areas are appropriate for your level and Jame’s leveling guides. I don’t follow the leveling guides religiously but I ready them for tips and a general idea of where I should be looking on heading next zone-wise. DwarfPriest’s levelling guide is excellent, but in need of a little updating, post 3.0.2 (as are most guides).
4. Don’t forget your buffs. Priests get a couple of useful buffs which they should maintain on themselves at all times when leveling. In particular Power Word: Fortitude at level 2 and Inner Fire at level 12. You can often get other free buff off other players by casting your Power Word: Fortitude on them.
5. Start your professions early. Regardless which you choose they are generally easier to at least partially level as you do. Unless your rolling in cash and starting an alt I suggest you choose at least one gathering profession. Herbalism, Mining and Skinning are all good money makers, but I prefer Herbalism of the three.
Blacksmithing and Leatherworking are both probably not good choices for priests as you can’t use most of the stuff you produce. Enchanting, Inscription, Tailoring, Engineering and Alchemy are all decent options for priests.
My priest Turkelife choose Enchanting and Tailoring. She has required cash from my other toons to level these (especially tailoring). Personally i’d recommend Enchanting as a useful leveling profession. You can enchant your own gear and its not overly onerous to level with greens you find and make from your other profession. Tailoring has had pretty much no benefit to Turklife while leveling other than providing more green items for her to level her enchanting with. Maybe i’m doing it wrong?
You First aid and Cooking secondary professions can both be levelled easily as you level. Fishing I believe is actually easier to level once your are high level. Fishing and Cooking can be levelled nicely together if needed.
6. Google for more. There are a lot of levelling guides both priest specific and general ones. Get off your butt and search for ’em.