While the same is not true for healers in the raid finder, you're the only one of your kind in 5-man content, where you're likely to start to learn to heal. Healing's a little tricky to get started with, just because there's not much scope to practice out in the world, without other players. I would definitely have a go with all your spells out in the wild, just to see what they do, but it's only going to be when you get into an instance that your skills are actually tested.
Starting as you mean to continue
One of the things that's going to affect your gameplay and learning experience as a healer is how you're getting started. Leveling a new healer from scratch is a great way to go, as the low level dungeons are pretty well stress free, and you'll get your abilities little by little, so you can learn how each one works and, just as importantly, weaves in with the other spells at your disposal.
If you're going straight in at max level, then you've got all your spells at once. It's a touch more tricky! Do check out the core spells in the spellbook's new tab to see just what you're dealing with, it won't explain how to heal with everything you've got, but it's a good start.
If you can, run 5-man normal dungeons, or easy heroics, with friends or guildmates. The best will be if you can get someone in your group who plays your healing class too, and they can give you tips on healing in certain situations. For now, I'd recommend grabbing all your abilities and dragging them straight onto your bars from day one. You won't use all of them all the time, but it's a good point to start from.
There is no rotation
If you're coming from a DPS role, you may well be wondering about your rotation, you know, the order in which you cast spells to get the maximum healing output from your spec. Unfortunately, healing isn't quite like that. The best spell to cast at any given time is the spell that is the best one to cast! Yes, there are some elements that are predictable, your class of choice may have a set of spells to use for AoE healing, for example. But, largely, it's a case of knowing what's in your toolkit and being aware of when to use it.
What's more, it's not just the type of damage flying around that you've got to consider. That's pretty obvious, really, and will differ from class to class. There will be times when it's better to use a few single-target spells, and times when it's better to use an AoE, and the best place to find this out is in class guides. Often there's a minimum number of people to be affected by an AoE heal or suchlike. But, apart from those sorts of things, there's no real rotation. It's a case of being adaptable, and knowing what tools you have for every situation.
Don't set too much store in healing meters
Our very own Dawn Moore wrote an article a while back about how to top the healing meters as a discipline priest. While it related to discipline in Cataclysm, the message is the same, snipe heals to get ahead. Sniping heals, though, isn't OK. It's not nice, it's not being a team player. If you're part of a healing team, rather than a solo healer, and you're with friends, then you should work together to put as little of your blue bars into everyone's green bars as possible. The way you get to be top of the healing meters isn't necessarily the most mana-efficient.
Healing Per Second isn't everything. There's no point, if you're a priest, in flash healing away your entire mana bar then hitting the report total healing button. HPS in that instance means very little! For starters, what you're after is effective healing, not just raw throughput. There's no point smashing out 100k heals into players who only need 50k to fill up their health bars. Only 50k of that 100k is effective. Efficiency is something to monitor on meters, some classes have it harder than others, but watch your overhealing, and try to keep that low. There's an overhealing measure available on most meters.
Keep your party under control
It's nigh impossible to heal people when they're not in range. It's also remarkable how often players forget this! You really do need to have your healer in range before pulling, particularly if there's a door that closes! Protip for behind closed door healing, though, if you're a shaman your totems and Healing Rain go through doors, as does priests' Prayer of Healing. I'm not exactly sure on other classes, but indirect heals are probably good!
So, if you're drinking for mana, say so. And don't be afraid to ask for mana breaks, you need mana in order to heal, and if a tank is going hell for leather through a dungeon with you chasing frantically behind them with all but no mana, that's not going to optimize your healing if it all goes wrong. Most healers have mana cooldowns, but not all of them get those cooldowns particularly early on, if you're starting a new character to heal. Learn what they are, and use them, but don't be afraid to stop for a drink!
Be aware of your utility
Speaking of cooldowns, while I don't play every healer, do also be aware of your utility. If you're PvP healing, this is particularly important, as you'll be more to your team than green numbers. To be a good PvP healer, you need to know what else you can bring to the table outside of just pure heals. Slows, interrupts, crowd control, purges, all these are examples of utility.
And it's not just for PvP either, utility is very handy for PvE in certain situations, particularly those where it all goes horribly wrong. Aggro drops, slows, roots, cyclones, other CC, stealths, fades, all those sorts of things can save your behind when things get out of hand in a dungeon just as much as they can in PvP situations.
What's more, be aware of your throughput cooldowns, your OMG cooldowns, your big party trick that saves the whole situation. Always take a second to bind new spells so they don't just sit languishing in your spellbook!
Consider healing addons
There will inevitably be a slew of comments saying "you don't need healing addons, I've healed since 1974 with just the standard UI." True, you don't need them, but they might make your life easier. I know when I first started healing, healbot made my life so much more straightforward. I now heal with a mixture of target healing (the standard UI method), click healing, and macros.
Neither one is inherently better or worse than any other, although I'd encourage something other than click healing for arenas, as it ties up your mouse, which you should really be using for movement. Nonetheless, if you like to click heal in arena, more power to you.
If you're finding target healing a bit of a struggle, I wrote a two-part series on raid healing addons, so you can make an informed choice! Do try them out, they might make your life a whole lot easier! Or, you might hate them. Everyone's different!
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to learning how to tank, getting up to speed for heroics and even how to win Tol Barad.