A few things:
I've leveled a Mage, Paladin, Druid, and DK (In that order) and each of them was their own experience. Coming from the mage world, I was very surprised to see the durability of the cloth wearing priest. At lower levels - prior to getting some of the self healing stuff - this isn't purely true, but generally speaking, priests don't play like other cloth wearing classes. Also as a mage/paly guy, I'm very used to the "AOE Grinding" and as a priest - you have essentially zero AOE abilities, but when you get into about your 60s, you can actually Dot-Up multiple mobs, shielding youself and allowing them to hit you (in cloth!) while they slowly die. It's not like my Paly pulling 10+ level 80 (Reg) mobs and grinding them down... But at 3-4 mobs - taken at the right time/place.... You can do it...
Here's what you want to do:
Stack the hell out of spirit and INT. The first tier of the Shadow spec has an ability that increases your spirit upon killing a mob. So - your goal is to DoT, Smite, Mind Blast, or Mind Flay things to hurt them, but then, stop casting and wand them to death. You'll basically do this until you hit outlands. In your 40s (I think) you'll get Shadow Form - which will help, but that's really the leveling experience of a priest. "DoT, DoT, CAST, wand........." *I am actually surprised that Blizzard has made it this way/allows it to continue.... It's really the most boring leveling I have experienced. Mage sucked too - so perhaps it's just the life of a caster to be mana whores.... Paly was incredibly fun and Cat-form druid was cool. But priest is just "DoT DoT Cast (Conserve mana) Wand to death.
Even now - at 71, my rotation is pretty simply - Vamperic Touch, SW:P, Mind Blast, Dev. Plague - Then want till dead - Or, if I want to restart my mana regen I recast Mind Blast (You'll learn all this in time - At higher levels you get a spell called Vamperic Touch which will give you a 15 second buff to regan mana upon casting Mind Blast) I can go about 10 fights without having a drink.
Also - remember - as much a people want to tell you what they did and how great their classes are - Most people only spent a few hours of Real Life time at those levels. They quickly forget about the lack of spells/skills at that point. (For example - you don't get SW:Death until something in the 60s I think - and the spell is generally not that helpful - Even then) So - having said that - Leveling sucks. No one likes it. You're best Time per level experience will be by grinding out quests you have already done and which you quasi remember. The only thing you need to do - is do it.