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.