Before you really get into leveling your new priest, there are a few tips I can give you to make the (fairly painful) process easier. Some of these things you may already know, particularly if you've played a caster class before. But if you've stuck mostly to melee, it will help you a lot to understand some basics about mana and mana regeneration. A lot of the tips here will help all casters, not just priests!
Spirit vs. Mp5 - Mp5 is pretty tough to get at very low levels. If you are low enough level to not have Meditation, however, pay attention to the difference. While both stats directly affect your mana regeneration, spirit only affects your mana regeneration when not casting.Mana per five seconds (Mp5) increases your mana regeneration both while casting AND while not casting. This is why at low levels Mp5 is much preferred, and much more difficult to get.
"While Casting" and "Out of Five Second Rule" (OO5SR) - "While casting", your mana regeneration stops, unless you have a specific talent, ability, buff, or Mp5. However, the definition of "While Casting" is not actually while you are literally casting a spell. "While Casting" begins as soon as your cast ends, and lasts for five seconds. So if you cast Smite, or even instant spells like Power Word: Fortitude, your mana regeneration halts for five seconds. If you cast one spell right after another, you will stay within that 5 second rule, and not regenerate any mana. That's why it's best to bunch your spells together, so that you can take advantage of as much time Outside of the Five Second Rule (OO5SR) as possible.
Until you pick up the talents Meditation or Spiritual Guidance, your stat priority is Intellect -> Spirit -> Stamina -> Spellpower. Intellect is especially important at low levels because Priests never seem to have enough mana to get through a fight - let alone heal themselves to survive! They need all the mana they can get. You aren't going to stop casting long enough to regenerate your mana while in combat unless you use your wand a lot.
Get a wand! I cannot stress this enough. Enchanters make a pretty decent wand that you can start using at level 5, and since they make them to level up, they are usually fairly cheap. Save up your coppers and silvers in those first few levels, and as soon as you hit level 5, run to the nearest auction house and buy one. You will save yourself a lot of mana, and a lot of deaths by making use of a wand for about the last quarter or third of a mob's health.
Stock up on health and mana potions. Whenever you can get them cheap on the AH, buy a ton of them. Now that they stack in lots of 20, it doesn't use much bag space - but you will be thankful for them, trust me. Use them liberally to make the leveling process much less painful. People always think Priests have it easy because we can heal ourselves - but at low levels, we just don't have the mana.
Stock up on Scrolls of Intellect - Scribes make these, and they drop all over the place, so you can usually get them pretty cheap. If you find a good deal, stock up. If you can't get scrolls, beg mages for their Intellect buff - I always buff Mages with Stamina, in hopes that they will return the favor.
Always buy at least one full stack of the highest level drink you can use - if not two stacks! Unless you run around with a Mage, you will go through those drinks very quickly. You don't really need to buy food, since you can heal yourself between battles, and then drink up to recover mana.
Spirit Tap - once you're high enough level to have picked up Spirit Tap, you may notice that after you finish combat, both your health and mana regenerate faster. Spirit also boosts your health regeneration, as long as you are not in combat - so you should find that after speccing into ST, you have a lot less downtime between fights.
Professions for Leveling Priest
Some professions are more helpful for leveling your priest than others. I know a lot of people pick up professions simply for $$$ but I actually like mine to be useful. At low levels, Priests need all the buffs they can get - especially Intellect. I recommend picking up a profession that will somehow provide you with relevant buffs.
Alchemy & Herbalism - Lifeblood from Herbalism is a life and mana saver. Level it up with health and mana potions, and when you are able to make Elixir of Wisdom, always use it.
Enchanting - Enchanting is not always easy to level, but coupled with a profession like Tailoring, you will have plenty of things to disenchant for mats, as well as the cheap green items you'll pick up along the way, and quest rewards that are otherwise useless. You can make very nice wands at low skill levels, and even enchants like Minor Mana make a big difference at low levels. Not to mentionWizard and Mana oils!
Tailoring - You can couple this with Enchanting and it works very well, since Tailoring is the only armor crafting profession that requires no gathering profession for materials. Instead of using cloth for first aid (and since you can rarely use bandages while in combat, and you can heal yourself while out, who needs it?) use it to craft your own nice armor - and BAGS! If you don't want to take Enchanting with this, Skinning is good because you often need some leather with tailoring. If instead you want a profession from which to make more money, any of the other gathering professions work well.
Inscription & Herbalism - Make your own glyphs, your own off-hand items, and scrolls. Scrolls of Intellect are very very handy, particularly at low levels. As mentioned before, Herbalism is also nice for Lifeblood.
I'm leveling up a new priest specifically to figure out what helps most with the leveling process. As I find new tips, I'll add them here. If you have any tips, please comment and let me know so that I can add them!