Sticky. Up top. Tells you how to heal.
Read it. Know it. Live it.
Meanwhile, at low level when you have no spells, keep Renew rolling on the tank all the time, and, pretty much, Heal for big damage (tank) and Flash Heal for little damage (DPS). It's good to Shield the tank too but until it gets talented it can be a mana hog, so at low level you don't necessarily want to do it every time. Then every time you get a new spell while leveling, learn what it does and why and when to use it, and start using it. The strength of a priest for healing is the huge number of tools in the box. When you hit level 40 you'll get Greater Heal and then you'll take Heal off your toolbar forever. Also make sure you have a unit frame that does something special (a border or such) when someone's got a debuff you can dispel and don't forget to keep up with that.
While leveling through instances is also a good time to try various healing UI's (Healbot, Grid, Clique, VuhDo, etc.) to see what you're comfortable with. We have lots of UI discussions here so I'm sure you can find plenty of info. I think most healers are still using Grid. Dadanox and a growing number of his fans like VuhDo. I just use Pitbull frames with Clique and macros, but that does not mean I am not Dadanox's fan.
As Aethien says, if you are planning on doing lots of instances on the way up, you may want to level with a Smite spec. I'd do 3/3 Spirit Tap first, then Holy up to Searing Light, then Disc up to Meditation, then finish straight up either up Holy or Disc, whichever tickles your fancy. If you decide on Holy and you're just questing, with Spirit Tap you may find you're never drinking anyway and in that case if you don't want to take a detour for Meditation that's fine too. If you're doing battlegrounds while leveling, you need Meditation because Spirit Tap will not kick in enough.
There is no reason that you can't heal instances as Shadow while leveling, and Shadow can be a very fun spec, but if you're mainly interested in healing it is worth it to get to know the healing trees well, and they are not inefficient for leveling anymore. They're also nice if you plan on doing battlegrounds while leveling. It's a lot of fun to heal battlegrounds when you're at x1 or x2 - DPSers can still get one-shot coming in so low but healers are always strong contributors regardless of level.
Edited because I can't spell on Sunday.
Edited, Dec 20th 2009 8:27am by teacake