You don't say whether you are discipline or holy, so i will confine myself to more general rules. I have all six specs of healer at 85.
1. Your job is to keep yourself and the tank alive. You are a cloth wearer. You die easily. Watch your own health carefully. As a priest, stay back! There are a few fights where you need to be near the tank, but those will be obvious and they are rare. Your spells cannot go through walls or around corners, so watch your line of sight.
2. If a DPS is determined to die, they are going to die. Do your best to keep all the dps and pets alive, but YOU are your first priority and the tank is your second priority. Do not use up all your mana on a DPS who is standing in purple stuff.
3. If a mob attacks you, run to the tank. Do NOT try to run away.
4. You do not have to use a raid frame add on like healbot or vuhdo, but it makes you faster. Sometimes on the forums people will ask "Should i use healbot or mouse over macros?" This is somewhat like asking if you want cereal or milk for breakfast. They work best together. They are NOT mutually exclusive.
5. Grow a thick skin. People do not like to accept that they died because they stood in nasty purple stuff, and will blame you. If you need lots of praise and hugs, healer is the wrong role for you.
6. Always do your best, even for characters who are rude or thoughtless. This next part is my opinion, and not everyone agrees, but... To me, if you are letting people die because you are mad at them, or because they are stupid, you should not be a healer. We are a support role, not the star.
7. Consider running each instance first as DPS before healing it. Watch for damage spikes and moments when large numbers of adds appear. Then when you are healing, have a shield or a renew on the tank and perhaps melee DPS when those spikes are about to occur.
8. Healing is mostly paying attention. I am aware that some WOW players watch movies or tv or text while playing. Don't do it while healing. Healing is sometimes long slow periods of tedium followed by moments when total concentration and alertness are required.
9. Read all your spell tooltips carefully and make sure you have keybinds that you can reach quickly for your main spells. Since we now learn new spells without going to a trainer, make sure you watch your spell book carefully and think about each new spell when you get it.
10. Sometimes tanks are very aggressive and pull new groups before you are ready. Get used to it. If you are low on mana, drink as soon as you can. Bind a macro that casts Power Word: Shield on your tank (who should be your focus.) You can cast this while running. You can also buy or make Swiftness Potion, which gives you a speed burst and has let me save many a rash tank.