The things you need to be concerned with in an instance are your aggro, not having people die, and your manna.
Once you get to level 68, you will get Prayer of Mending, and it will make a good opening spell. This is because it gives aggro to the tank and not you. For now, you don't have that option.
Once the tank has engaged the enemy, give him a moment to establish aggro. Then throw renew on him. Keep renew on him the entire fight. Renew will slow the incoming damage, giving you time to throw flash or greater heal. Also use renew as a quick way to throw some heals out to others in the party, so you can go back to the tank. If you had PoM, you would throw that into the spell rotation as well.
Once you have started to heal, you may have adds head your way. Be ready with fade, to decrease your aggro and send them back to the tank. My healing is high enough that I actually gave up a bit of +healing to carry a Timelapse Shard trinket as a second aggro reducing spell.
Flash heal is quicker and uses less mana than greater heal, but heals for less and is less mana efficient. Some people try to mostly use greater heal for the mana efficiency. It depends on the situation. I use both.
Downranking is a way to conserve mana, but Blizzard is soon making changes to eliminate its advantages. I wouldn't bother with it at this point.
Your AOE heals are Prayer of Healing and Circle of Healing (if you have the talent). Some mobs do aoe damage to the entire group, or the entire group may be down in health. These heals are great for those occations.
Binding Heal will heal you and the target. It's great when you need to heal fast, but need heals yourself too. There are certain occasions (Nightbane's skeletons) where I basically spam this on everybody as they all take damage.
Beyond this, you should be ready to Dispel magic and disease, and shackle undead.
You need to be able to use all of these abilities quick. I especially recommend keybinding fade and shackle undead (use a
focus macro). These you will want to be able to hit fast without even moving the mouse.
When you start raiding in larger groups, you should look at addons that provide you the information you need, like
Grid or Healbot. Whether you click heal with your mouse (Healbot or Grid with Clique) or select with the mouse and use keybindings to heal (what I do using grid), you want to be able to heal quickly. Selecting a target with your mouse, then mousing to the spell on your taskbar, then back to another target, then back to the spell bar isn't going to cut it for long.
One last comment, don't shield the tank. Tanks need to gain aggro. Only use your shield on the tank if you need to have time to throw a heal or he will die.
Just get in there and practice. Priests are a very versatile class with many healing abilities. Each lends itself to a different situation. Have fun, and good luck in that instance.