This is less a comprehensive guide than it is a more general set of guidelines.
Leveling Spec: Discipline
This isn't the typical raiding spec for discipline priests. Your tree should look something like this once you reach the max level. You get Penance at level 10 once you unlock the ability to use your talent tree. Prioritize points into anything that supplements damage, like Atonement, Archangel and Evangelism. For kicks, I added Reflective Shield, since I'll be doing questing; if I'm going to be hit, I may as well turn some of that damage back onto them. Once you get Power Infusion, remember to use it whenever it is available to boost your DPS throughput. Don't forget that Penance can be used offensively against targets. While it is typically used for healing, just know that the option is there if you need a finishing spell.
Leveling Spec: Holy
The spec is almost the same as the traditional raiding holy spec, with a few changes. Unlike the discipline spec, you're banking on extra healing power for additional survivability. You may as well learn how to use your Holy Word spells early as you're leveling up, anyway, since you'll be relying on them at the endgame level.
The best spells to use
Ready to tackle Azeroth's biggest threats? They'll probably be boars of some sort. Make sure you start out as far as possible from your target. You want to maximize the amount of time it takes for a mob to get to you. As I'm engaging mobs individually, I like to open up with a Holy Fire first to get the DOT going. I'll follow that up with a Shadow Word: Pain, then continue to repeatedly Smite until I can move on to the next one.
If tackling multiple mobs at once, apply Shadow Word: Pain across the board. Keep casting Holy Fires and Smite on each target, one by one. If they close within range of you, keep your Power Word: Shield and Renew up. Make sure your DOTs stay refreshed. At this point, your goal is to try to survive the incoming damage just long enough for your spells to take them down.
In dungeons, your spell use is obviously going to vary based on your level. You unlock different spells and abilities as you level up, and you'll have a limited set of tools to use at lower levels. As you gain levels, you'll unlock new spells to add to your toolbox.
Level 1 Smite
Level 3 Flash Heal
Level 5 Power Word: Shield
Level 8 Renew
Level 10 Penance and Holy Word: Chastise
Level 16 Heal
Level 18 Holy Fire
Level 38 Greater Heal
Level 44 Prayer of Healing
Level 48 Binding Heal
Level 68 Prayer of Mending
Don't worry about using Heal at lower levels. Stick to Flash Heal and Greater Heal. Keep Power Word: Fortitude up and Inner Fire up!
Why use Renew? When I'm coaching new priests, some of the players ask me why should they use Renew when a spell like Flash Heal is enough to get players back over the top. I tell them that in different encounters, there will always be different people taking damage. Your Renew gradually heals players back up to safe levels. It is an instant-cast spell, a fire-and-forget. So what I tell players to do is to cast it on themselves or a different DPS player (generally a player who does not take constant damage). Single-target healing spells are usually for players who are going to be taking damage for some time. This way, your main healing efforts are focused on players who are under immediate threat, and everyone else is still receiving heals from Renew.
What are my stats? Intellect is your primary stat. It increases your mana pool and your output for healing and damage. As you get to higher levels, you'll want to slowly add some spirit to your gear. Secondary stats like haste and crit aren't a high priority while leveling up, but they're nice to have, as they'll increase the chances for your spells to hit harder and allow you to cast your spells faster.
Which is better for healing while leveling, discipline or holy? I personally like what discipline has to offer because of durability it has. It offers the best of both in terms of healing and survivability. I'd consider making a second spec holy just so you can fiddle around and get used to the Chakra mechanics.
What race is the best? For Horde, I'd suggest blood elves or trolls. Blood elves can Arcane Torrent; trolls can use Berserking. For Alliance, I lean toward humans (The Human Spirit) and gnomes (Expansive Mind). Dwarf priests will always have a special place for me because of Stoneform.
Good luck on your journey from 1 to 85!
Spiritual Guidance has the inside line on pre-raid gear, valor point and raid gear, and healing strategies for bosses such as Atramedes and Chimaeron. Newcomer to the priest class? Look into Discipline Priest 101 and Holy Priest 101.