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.
First off: I would warn against basing any decision like this on month to month balance considerations. Things change so often in WoW that you should really decide based on how much you enjoy a particular classes playstyle, toolbox, and yes, even aesthetics. Healing Balance is Pretty Good right now. All five healing specs have a role to play, and a (25 man) raid which doesn't have the flexibility to see them all represented is a 25 man raid that is going to run into trouble. At the five man level, all five specs are more than capable of handling any content you'll see, though Discipline Priests in particular may find Heroics somewhat more challenging than most due to increased gear dependency.
That said, the current state of PvE healer balance at the high end is this: Priests are somewhat above average at Group and Raid healing - this is reflected by the nerfs to Prayer of Healing in the upcoming 4.0.6 patch, but those nerfs don't really change the overall state of things. Additionally, Priests bring extremely powerful cooldowns to a group such as Pain Suppression, Guardian Spirit, Leap of Faith, Divine Hymn, and Power Word: Barrier, all of which are defining abilities that have had entire strategies built around them. If you see raids specifically looking for Priests to heal, this is why.
Shaman by contrast, are currently the kings of PvP healing because they are extremely durable, mobile thanks to the new Spiritwalker's Grace ability, and bring substantially more offensive utility than other healers thanks to Heroism/Bloodlust, Wind Shear, Totems and Purge. In PvE, every 25 man raid wants at least one Resto Shaman for Mana Tide - and 10 mans love to have it as well. Mana Tide is the single most powerful mana regen ability in the game, and it is absolutely essential for success for many groups in many encounters. Additionally, you may find it easier to gear up a Shaman for healing because there will be less competition on mail pieces with intellect than you will find on similar cloth pieces - and because if you already have decent DPS gear, you always have the option of DPSing your way to a healing set.
My advice to you? Try out healing on the shaman. Chain Heal, Spiritwalkers Grace, Riptide and Healing Rain are all enormously fun abilities, and the Shaman healing style is unique among healers in WoW. Unless you're pushing bleeding edge content the class balance issues seen there are going to be largely moot for you - what matters much more is your own competence and gear.
As an aside, checking the WoW forums will always find you nothing but gripes. A saying I'm fond of is that the forums are as much an indicator of players happiness as a hospital is an indicator of public health.
Best Answer
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).
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.
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.
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.