First Time PriestFollow

Look at your keyboard.
There is a button on there that says 'enter'.
You should use it.

You should also probably look at the threads on the first page of this forum.

You can throw healing spells even though you level as shadow. Simply don't go into shadow form when your role in a group is to heal. At 40, if you want to go for a dedicated healing build, but keep shadow for leveling, get dual spec.

What ever you do, keep spirit tap until you are done leveling. That way you won't have to drink as much. Once you get a few levels behind you, you can start looking at popular builds.

Personally, I wouldn't try to be a holy / shadow hybrid. Especially at first, you aren't going to have enough talent points to do anything that is very effective. It's better to beef up your main spec as best you can.

As Dadanox suggests, I would not level with a Holy/Shadow build. You can, however, level in any of the three trees. Although most people find Shadow most efficient, there are plenty who swear by Holy or Disc as well.

I also echo the advice for putting the first three points in Spirit Tap no matter what else you do. It is the best way to level without having to stop to drink.

While I don't disagree with Moz that hybrid builds can work well for Disc, if it's your first time playing the class (or the game!) I personally would recommend picking a tree and sticking with it for now. I think getting up to some of those tree-defining talents at the top is a must for learning what that spec is really about and how the play style works. This is not to say you'll never put points in another tree (Spirit Tap regardless of spec is a good example), but overall I'd put the vast bulk of your points into your tree of choice.

And as Aethien mentioned in his own special Aethien way, there are some "sticky" threads at the top of the forum that you should read through.

Also always have the best wand you can get off of the AH for your level.

If you can end with the wand then the spirit tap will return more mana each time.

Fast is a relative term. If you merely want to speed through the process, I would suggest following a levelling guide. I followed Azuarc's guide. It is a little dated now, but still will suit your needs.

As far as builds and gear, I'm going to go out on a limb and say it doesn't matter what you put it in until around L40, and then I would suggest respeccing into one tree with the goal of reaching the top tier talents of one. Sure, Spirit Tap will keep your uptime higher, but most of the early talents only complement the skills you should be developing as you level. (If your buddies are willing, let them buy you dual spec so you can switch between specs for added experience.)

Knowing what to do and when to do it is key for a priest. You will begin to accummulate new spells/abilities as you level. Learn to use them in their proper context. Would it make sense to toss SW:P on a target about to die? Of course not.

If your aim is to level fast, then you need to learn how to survive, above all else. You will learn rather quickly how much you can take on without breaking your rhythm. One day, you will come across a three or four pull that you would have normally passed up for easier prey. On that day, you are going to throw caution to the wind and pull them. If you walk into it trying to burn them all down, one by one, you will probably die. But if you devise a plan that uses your wide ranging abilities of attack, defend and prolong, then even if you do die, you take a step towards developing the skill that good priests have.

Oh and one more thing...

Never... no no, that's not right.
Always make sure you look good doing it. Even if you have to drop 25g on a doublet, make sure you are dressed for success.