Thanks to @tzenes for an ElitistJerks link, which has some great info. Of note is that Shadow Word: Death is much better than it used to be.
Summary:
- No specific information on how to start the fight
- Keep up Vampiric Touch, Devouring Plague, and Shadow Word: Pain
- Shadow Word: Death when available
- Mind Blast when available
- Mind Flay for filler
One important note is that (with a Glyph) Shadow Word: Death has an instant cooldown when the boss is below 25%. This means you'll need relatively little Mind Flay filler past the 25% mark. Just don't kill yourself.
I am going to suspect that Mind Blast could eventually outperform Shadow Word: Death later on, especially with Mastery points making it more effective and Mind Spike synergy. But according to Elitist Jerks, Mastery is pretty terrible at the moment.
The short explanation is that you stay in Shadow Form 95% of the time, throw DoTs on everything, then sit back and watch the enemy's face melt. Remember that you can still cast Discipline spells, namely Power Word: Shield. The only reason to leave Shadow Form is to cast an emergency heal (or three, since they're going to be pretty wimpy as a Shadow priest).
DoTs
You learn three main damage-over-time spells, which are pretty self-explanatory:
If the enemy's face isn't melting fast enough for you, you can use Mind Flay or Mind Blast to speed things up. Once its health drops below 25%, finish it off with Shadow Word: Death.
You should be able to easily juggle multiple enemies at a time by casting DoTs on everything, then using Psychic Scream (or Dispersion) to keep them off you until they die. For me, this worked all the way through to level 80 where mobs started to get beefier.
Mana
Shadow Priests get a lot of mana replenishment abilities.
You'll definitely want to use Glyph of Spirit Tap, which replenishes mana when you kill an enemy using Shadow Word: Death, and Masochism/Glyph of Shadow Word: Death which replenishes mana when you don't kill an enemy using Shadow Word: Death.
Eventually you'll learn Shadowfiend, a temporary pet that gives you mana when it attacks, and Dispersion, which replenishes a bunch of mana and prevents 90% of damage for a few seconds.
Mind Spike
Once you hit level 81 you'll learn Mind Spike. Along with the Mind Melt talent, this gives you an alternate way to open combat: Instead of applying DoTs, you can cast Mind Spike twice then follow up with an instant-cast, guaranteed-critical Mind Blast which (with Paralysis) stuns the target for 4 seconds. If the mob isn't already below 25% health after that assault, you can just throw out another Spike or a Mind Flay before finishing it off with Shadow Word: Death.
Best Answer
You should be fine healing in Shadow Specialisation at low levels, especially if you're only using Shadow Spec until you get Dual Specialisation, which is level 30 or so now.
At those levels you should have no problems at all keeping the party up.
That being said, up until Dual Specialisation level Discipline does perfectly sufficient DPS and would actually be quicker to level with because of the mana issues you suffer at lower levels as Shadow.
Either way you choose to go, the way you'd go about it would simply be keep Power Word: Shield up on the tank as much as possible. For healing it depends if you choose to stick with Shadow or go with Discipline, but as Shadow you will be using Flash Heal and as Discipline you will be going with Penance.