It has been quite the experience doing more and more healing from a mainly tanking and dps background and I noticed a few things along the way that helped me do better.
1. Stop doing damage. I can't say how much healing as a tree has improved my healing as a shaman. Since trees can't do any damage aside from some humorous branch swinging animations it really takes your mind off damage. Even if you can put out damage while healing you really shouldn't. You may have extra mana now, but you might need it in 10 seconds. Don't be the healer that misses a heal because they were DPSing, we mock a guild member to this day for that. That said, once you've got a handle on your healing; go ahead be naughty and do some DPS, it's fun!
2. Watch your own health. I like most have my player frame larger and above my party. This make it really easy to overlook while watching other people's health plummet. Not so much an issue in raids since healers look out for one another. I toyed with displaying a party frame in addition to my player frame for 5 mans, but I decided to just be more cognizant of my health. Shamans have a nice fallback when they neglect themselves which has saved my butt more than once, rest of the healing classes have to be more careful. I'm sensing a trend that shamans are not the best choice for a first time healer.
3. Triage. There will be times you can't heal everyone, those are the times I find healing the most fun. You are the first priority, put the mask on yourself before assisting others. If the healer goes down so does everyone else. After that the tank is slightly higher priority than the DPS. Tanks have quite a few tricks to stay alive if you ignore them for a little bit to heal someone else. DPS usually don't have that luxury. DPS will also usually be taking less damage so taking the time to get a quick heal off on them will keep them in the fight and it will probably only take one or two heals. Don't feel bad about letting someone die to save the group though, in fact don't think twice about it. In those sorts of situations where split second decisions are needed there is no time to dwell on one death, concentrate on healing everyone else.
4. Let them squirm. Healers are all about positive reinforcement, you do something good and you get a heal. We don't want to reinforce a negative behavior after all so if someone is constantly pulling off the tank and/or taking unnecessary damage then don't reward them. I've only withheld heals once in my healing career and it worked great to deter pulling aggro. Keep in mind that it is a whole lot of wasted mana/time to rez someone so it's best to keep them alive, but not much more than that. We wish to help the group move along after all.
5. "Mana"gement. You don't want to make everyone wait every pull for you to drink back to full mana and you also don't want to have the group wipe because you had none. Make use of mana regeneration abilities often keeping them available for boss fights whenever possible. Drink between pulls, even chain pulling at a brisk pace there is often enough time before the tank needs heals again to get a quick drink if you hang back a little. If you do need more time let the tank know since tanks aren't always looking as closely to mana as healers might like. In a tight spot be creative, use bandages; I find the situations where I need to make every drop of mana count the most enjoyable.
6. Avoid healing aggro. Easy coming from a tanking/DPS background right? I know I've been one shotted healing before doing just that. Don't heal your tank right away after a pull, give them a chance to build aggro. Healing aggro is very small in comparison to DPS so you can start healing pretty quickly.
That's about all I can think off the top of my head. I did quite a bit of healing over the weekend in PUGs with a mixture of skill levels. Healed tanks that I would describe as naked mages in terms of mitigation in addition to experienced geared tanks that hardly needed any healing. Is it wrong for me to enjoy the former more? The well geared tanks make up for it in speed though. I was doing an AN run with a very good tank and he pulled the last boss when I had no mana; ok I had like 500, but still. Of course we did fine, that fight is crazy easy to heal as a shaman. I like to think the tank knew that and was making it fun for me.