Hey, the God Emperor of Hunters is here, too. He might tell you to play druid, though.
Firstly, you don't have to solo. Starting at level 10, you can enter PvP battlegrounds and heal the peeps there (although it is advisable to start at level 17 or so because all the level 19s will crush you), you get put into a group with 9 other random people and the queues aren't that long.
More importantly, at level 15 you can use the Random Dungeon tool and - especially as a healer - quickly find dungeon groups, again with random people, often not even from your server.
Finally, priests are great to solo. You can specialise your character through talent points, and you have the option to spec Shadow which limits your healing capabilities a bit, especially at higher levels, but makes for great soloing/damage dealing, or you can go for Smite spec (which we all love here), which is a healing spec at its core but takes some talents that allow you to deal decent Holy damage. If you're interested in that, you can ask and 3 people will tell you how to spec like that.
Priests in WoW are easier to solo with than anything but maybe BST in FFXI.
Also, there are other healing classes in the game, being Paladin, Druid and Shaman. I've got one of each, although my paladin isn't really at any level where she'd heal yet.
I personally dislike Shaman healing as it is right now, but it used to be nearly as awesome as priests, and tastes might differ. I've come to like druid healing. My druid is only level 29, but at earlier levels you can just go into Cat form to deal damage even if you're specced for healing.
At level 40 you have the option to get Dual Spec, allowing you to have two talent specialisations you can switch between, so you can go for a healing spec and a damage dealing spec if you want. It costs quite a bit of money, though.
If you want help choosing a healing class, here's a breakdown for you:
Paladins do single-target (or 2-target) healing and are quite amazing at that. They also only have 2 buttons they need to press for healing, pretty much.
Druids have many different healing-over-time effects and that's basically the key to their healing style. They don't do big numbers but do a fair bit of healing continuously.
Shamans have Chain Heal. It heals like 3 targets, I think. They also have totems, which are cool. I think shamans can fill every healing role to a degree, but Chain Heal is their awesome spell.
Discipline Priests mostly mitigate damage before it comes in instead of healing it, and also excel at burst healing (doing more healing than anyone else on a single target for a short amount of time). They're traditionally single target healers, but shielding everything in sight can be very effective, too.
Holy priests are the best at group heals, but they can do a bit of everything else, as well, when needed.
Okay, I'm finished now.