1. Turn off your meters. It's hard, I know. But from my experience, most healers (or players for that matter) tend to focus too hard on meters to the exclusion of doing their job well. A good healer might be the bottom of the meters but they dispelled every debuff in the raid and they used their heals more effectively, rather than topping off a bunch of DPS who were only down by 1% or so.
2. Run PUGS. The worse the better, both 5 man and LFR. If you can keep a really bad group up through stuff that they shouldn't survive then you know you're a good healer. If you can't do that, then PUG some more! I hate bad PUGs as much as the next person, but they have made me a stronger healer. Plus, I get free goodie bags!
3. PVP. This is something I need to do more of. If you PVP you'll have a better understanding of what to do in a variety of situations. It's especially great because you can learn how to defend yourself in a tight situation, and then you won't panic when a dozen mobs start running towards you in the middle of a raid. You will also learn to think on your toes and incorporate spells you might not normally use in a raid setting.
4. Force yourself to change your healing style. If you're a disc priest, try running an LFR without using Atonement at all. If you're a monk, run an entire raid as ranged, or melee, whichever you are least comfortable with. Familiarizing yourself with methods of healing that you aren't comfortable with will absolutely pay off down the road.
5. Watch other healers of your class. Whether good or bad, you can see what they are doing and either avoid bad practices you see in others or adopt new things you see someone else doing. I learned something last week by watching a really strong discipline priest and now my throughput has shot way up, and I thought I knew everything!
6. Be open to new ideas. If someone comes to you and tells you you've geared all wrong and you should do something else, listen to them and try it out! It might not work for you, and they might be completely wrong, but if you don't try it you will never know. And even if it doesn't work for you now, there may be an encounter in the future where it will make the difference between victory and defeat.