As a healer (or worse, a tank) you need to get used to being kicked, yelled at, told how to play your class, and in general annoyed. Develop a thick skin, or just stay DPS.
Healing and tanking are both thankless jobs, and no matter how good you are, your party can screw it up. I've been healing since 1.0, and it never changes. I did a heroic random yesterday with a tank who I don't know how they passed the gear requirement for the instance, and three mediocre dps, and we only made it through because I was raid geared, and I flasked myself after the first wipe. I shifted to bear on the last boss and finished the last 10% by myself, and I still got shit. (If someone has crappy gear, and gets one-shotted, that's your fault, because you're a crappy healer, right?)
But I digress.
Before you start healing seriously, you need a couple of things:
- A good healer mod (HealBot or similar).
- A healer spec. Only priests have two valid heal trees. Everyone else needs to focus on Holy/Resto.
- Healer gear. Int is almost always the go-to skill now. Don't be afraid to use cloth gear if you have to: if the tank is doing their job, you shouldn't have to worry about soaking damage (if they're not, you're pretty much screwed anyway).
Earth shield on the tank, water shield for mana regen (shouldn't need it with the mana tide totem, but never underestimate a parties ability to ruin your mana pool), and always heal yourself first, the tank second, and the dps last: if you or the tank go down, you're pretty much screwed either way unless you have a backup tank lying around.
I wouldn't worry too much about healing until you can get dual specced: leveling with a healer spec is extremely annoying. You'd basically have to do nothing but run instances, and that's a very slow way to level. That definitely makes instances like the Deadmines annoying, but what can you do? You could probably heal it as enhance, if you were 6+ levels over everyone else, but that's not much fun.
And early on, "gear" isn't a big concern...Don't think you need some kind of super special gear. Just don't sell/disenchant the caster quest gear you get, and you'll be fine. Throw it in a bag, and put it in your bank, and dig it out when you want to do the healer thing. Don't be the lowest level guy in the instance, and you'll be fine. If you're worried about your mana, drink at every opportunity.