Ok as I have mentioned, I am currently in the process of leveling a new Shadow Priest. I thought I would share some really quick tips that I think are essential whilst leveling:
1. Get a wand - This should not be underrated. As a shadowpriest, you will have a lot of mana issues during the early levels and the entire process can be hell unless you have a backup. Get a wand as early as possible and maintain it. Possibly one of the most important items you will have whilst leveling.
2. Inner Fire - is awesome. Keep it up at all times or you will get pounded by any melee opposition. It has only 20 charges (unless you improve this in the discipline tree) and often needs to be refreshed.
3. Power Word Shield - is even more awesome. This spell is the difference between happy shadowpriests with full HP and unhappy shadowpriests running back to their corpse in spirit form.
4. You're a hybrid act like it! - You have the option to heal yourself at any time if needed. Don't get bogged down in your damage dealing and forget you can heal yourself or other party members and do a damn good job. This applies to questing and even more so to instances.
5. Psychic Scream - is a lifesaver. However use it as a last resort or if you are very confident of your surroundings. Fearing mobs can often create bigger problems when you accidentally aggro too many.
6. Dot and Run - You may look a little silly but you can easily kite mobs (or players) and maintain your instant cast DoTs on them. Let them chase you whilst you chuckle at their lack of HP. You usually need some space for this to work vs mobs but it is often a way of beating higher level or even elite mobs.
7. Mana is not your friend - Until you obtain spells such as Shadowfiend and Dispersion you will not have the best mana regen in the world. For this reason make sure you always have a stockpile of water and mana potions. They will make your life so much easier.
8. Spirit Tap - Probably one of the most important talents to pick. Get this straight away and don't be afraid to use gear that has plenty of spirit whilst leveling. The same goes for Improved Spirit Tap. Anything that gives you mana or makes you use less of it is a definite yes!
9. Use a threat meter - Whilst in low level dungeons a decent Shadowpriest can easily run into some threat issues during instance runs. Semi decent DPS + Group healing (Vampiric embrace) may land you in some trouble. A threat meter such as Omen is always handy.
10. Enjoy it - Shadow is a great spec and should therefore be enjoyed. Try questing , running instances or battlegrounds and make sure you do what feels fun. There may be parts where you find it tough but ultimately it will improve your gameplay when you overcome each challenge.