The first Instance you can do as Alliance is the Deadmines in Westfall. You should be at lvl 20 before you try and do that one.
As a priest, you should have NO trouble getting a group for an instance. The questing once you hit 15 kind of sucks in Westfall, because you get a couple quests that are easier in groups if you're at/below the quest level. My solution to that problem was to go quest in Loch Modan (another alliance 10-20 area). I got to level 17 and could solo the Defias quests that were a little difficult before.
Of course, if you're in Loch Modan, you could always try questing in Westfall for a while. Darkshore is also a lvl 10-20 alliance zone, but it's not as easy to get to as Westfall or Loch Modan and requires you run though contested (although fairly safe) lands.
After you've done the Deadmines (do it a couple times, some nice gear there, especially the Emberstone Staff), you can start thinking about doing Blackfathom Depths or The Stockades. I haven't done BFD yet, but the stockades is kind of underwhelming (fairly short though and there's a good number of quests).
Once you hit level 30-35, I'd check out Gnomeregan. Groups definatly need a Priest for that instance. It's a huge instance, I did it last night for the first time, we played for about an hour and a half before wiping near the end boss. We skipped probably 75% of the mobs there, just killing what we needed to get to the minibosses. The first time you play it, make sure you get to the boss that drops the Workshop key, that'll allow you to go in the back door of the instance and skip a ton of mobs. The end boss drops some great stuff, so I'll probably do this one several times as well.