Starting a Priest WoW
Priests are the game's most versatile healers, as evidenced by their two healing specializations--something unique to this class. Discipline priests are excellent in 5-man instances and are almost unkillable when playing solo, as well as valued PvP healers. Holy priests have access to a whole slew of area-of-effect and group-wide heals, making them in-demand for complex and difficult raid environments. And Shadow priests are now one of the most potent damage-dealing specs, literally melting the faces off their enemies while driving them insane.
Of course, it's not always easy: healing classes can be tricky to level (especially on PvP servers), and because priests don't get any multi-target attacks until quite high levels, starting off can be tricky.
Early instances as a Shadow priest
As a Shadow priest, your job is to do as much damage as possible. But because dungeons are complex environments with hundreds of enemies, it's extremely important that you follow your tank.
The tank is a player (normally a Paladin, Warrior, Monk or Druid) who wears lots and lots of armour--which means they can take a lot more damage than any priest. They also have special abilities which force enemies to attack them, rather than their weaker allies. (Like you!) Your tank will keep you safe, but only if you help them out.
In order to ensure that enemies are only attacking the tank, the tank will usually "pull" enemies themselves: they decide who the group will attack next, and will attract the target's attention by running straight at them. Do not attack the target until the tank has already picked them up, and try very hard not to accidentally "pull" extra enemies: focus your attacks on the tank's targets, not on anyone else nearby.
Early instances as a healer
When you run a dungeon as a healer, you are responsible for keeping everyone else alive. In general, you'll want to use [Power Word: Shield] on the tank (a player signified by a shield icon) in order to protect them from incoming damage. Only use your active heals ( [Flash Heal], [Renew]) to top off someone who has already taken damage, or to keep someone alive after a shield breaks. To cast a spell on another player, just click on them (or on their picture to the left of your screen) and cast the spell as if you're casting it on yourself.
More advanced Holy priests will eventually depend less on shields and more on sustained [Renew]--but for now, stick to shields. To help yourself conserve mana, you should also try to only cast healing spells: leave the attacking to the other 4 people in your party. Once you acquire [Purify], you can assist your allies even more by dispelling harmful effects that might otherwise endanger them.
If you enjoy running instances as a healer, you may find a healing add-on to be useful. The most popular add-ons include Healbot (which makes it much simpler to cast healing spells on your party members), Decursive (which will make your use of [Purify] more efficient) and Deadly Boss Mods (which will warn you when a boss or environmental danger is about to do a lot of damage, so you can get ready to heal hard and fast).