Wyrmrest Accord (revered): Ancestral Sinew Wristguards. You could pick these up shortly after hitting 80, if you spend some time in a Wyrmrest tabard. They're quite solidly itemized for a Holy priest, and the socket gives you some flexibility. You'll also want Wyrmrest revered for the Arcanum of Blissful Mending, our best option for a head enchant, and for...
Wyrmrest Accord (revered): Gavel of the Brewing Storm. There's no Spirit on it, but Spirit isn't as huge for us as it used to be anyway, and crit is much stronger due to the number of talents that feed off it that most of us have.
Horde Expedition/Alliance Vanguard (revered): Charged Wand of the Cleft/Gnomish Magician's Quill. If you do both starting zones, you'll have Exalted before you hit 80, so this should be no trouble. It's good enough, although there's a slightly better option from quests in Storm Peaks.
The Oracles (revered): Fishy Cinch. I don't mess much with the Sholazar factions, but if you do, this is a great belt.
There are also some rep epics to keep an eye on (Robes of Crackling Flame, Sandals of Crimson Fury, Signet of Hopeful Light), but since you probably won't get Exalted with anyone before you start on heroics, I'm not going to go into them here. The Sandals do give you one more reason to work on Wyrmrest rep first, though.
The quests for the various high-level instances almost universally have great items for rewards. This is excellent, because it means if we can get a group for an instance once, we're guaranteed at least the quest rewards, even if we get no drops. You should be able to put most of an outfit together from instance quest rewards alone. And the gear you got from leveling quests will be perfectly sufficient to heal these normal-mode instances.
Bauble-Woven Gown (from Junk in my Trunk). Again we see stam, int, spirit, and spellpower, some of my very favorite stats. I love that they're putting stam on all the gear we want anyway; no longer do I lead in deaths (that's usually the mages these days).
Cowl of the Vindictive Captain (from Vengeance Be Mine). It doesn't have a meta slot, which is too bad, because the Insightful Earthsiege Diamond is amazing. But it'll do just fine until you pick up a Forgotten Shadow Hood or something.
Culling of Stratholme:
Gloves of the Time Guardian (from A Royal Escort). Not much to say about these; stam/int/spi/SP again. The Spellweave ones are better, but these are much easier to obtain unless you have piles of gold sitting around.
Not my favorite instance. The last fight still feels clunky to me. The quest rewards aren't amazing either, so IMO you'd be justified in skipping this one.
Flourishing Band (from The Struggle Persists). Really more of a pally ring, but still perfectly decent.
Cuffs of Gratitude (from A Wing and A Prayer). No stam, but we like all the stats on it, so that's fair enough.
Halls of Lightning:
The home of the deadliest boss in WoW (really not so bad on normal mode).
Lightning Infused Mantle (from Diametrically Opposed). My only complaint about this is the color and model - haven't we moved past that teal by now? One more reason I wish we could hide shoulders.
The other quest in HoL, Whatever it Takes! (which I believe to be the only quest in WoW that requires a single tongue as an objective) gives an awesome Shadow robe, since it's loaded up with hit, but healers will be better served with the one from Pinnacle.
The Chilly Slobberknocker is from the Arena of Anguish, WotLK's answer to the Ring of Blood. Ask your guild to help you if you can't find a group; it's fun and easy for five people, and certainly doable with fewer. I was wielding this puppy straight until I got the War Mace from Heroic Nexus, mostly for want of a good off-hand to go with my Hammer of Wrenching Change (from an Icecrown quest).
Sirana Iceshriek/Changing the Wind's Course, in Storm Peaks, give the Wand of Chilled Renewal, which I must confess I am still equipping, after several Naxx clears (seen Contortion drop once, didn't win the roll).
The crafted items that are worth getting at 80 are also rather spendy, so read on if you have several thousand gold you're willing to spend on your priest. These items will stand up to Naxx gear.
Interestingly enough, in my opinion, the Moonshroud set is not the best one for us. You'd think it would be, given the example of Mooncloth and Primal Mooncloth, but when you line it up against Spellweave, I think that set comes out ahead (despite the fact that you'd expect it to be more mage-oriented). It's a trade of a boatload of Stamina, a touch of Int, and a fair chunk of Spi for a massive amount of Haste; I find the Haste useful in getting that Gheal off before someone dies in a heroic, rather than a fraction of a second after. But it's down to personal taste, really. Both Spellweave and Moonshroud are good choices.
There are also two Jewelcrafted pieces that serve very well, although they're quite expensive: the Titanium Spellshock Necklace and Titanium Spellshock Ring. The sockets give them versatility, and getting your crit up is nice.
Wispcloak is one of the better cloaks in the game for Holy priests right now, outside of raids, and isn't horribly expensive mats-wise. It is, however, rather ugly.
Finally, unlike BC, some of the BoE blues in the game right now are actually useful. Two that stand out particularly are the Runecaster's Mantle (a zone drop in heroic Utgarde Keep), which I'm still sporting, and the Prison Manifest, a zone drop in heroic Violet Hold. Both of those drop with some frequency, and you should be able to pick one up pretty cheaply.
And there you have it: some relatively easy to obtain gear to help you get started in heroics and raids. I'm sure I missed some obvious pieces, so let me know in the comments. Happy healing!