The biggest hurdle a newly 80 Priest must overcome is the lack of good gear. Through questing and a few dungeons you may have acquired a few decent pieces, but most raid groups are going to want you to have, at the very least, full level 80 blues. The best place to gear up pre-raid is in Heroics, but you may not even be geared enough to do them just yet. You could run some of the higher level normal dungeons, but those are usually rather difficult to find groups for. A better option is to seek out level 80 BoE blues and epics, and crafted items.
For example, the Titansteel Spellblade or Titansteel Guardian are fantastic epic weapons crafted by blacksmiths. A jewelcrafter can outfit you with rings and a neck, either epic or superior depending on the size of your bank account. If you're a tailor, you should focus on crafting one of the epic sets that suits your chosen spec.
Finding gear upgrades is as simple as visiting the Armory and using the "Find an Upgrade" feature. However, this feature will often not take into account your spec. That is why it important that you understand how to gear your character, so that you can recognize an upgrade on site.
Gearing a Holy Priest
Holy Priests rely heavily on critical strikes. Additionally, they gain necessary bonuses from having high Spirit. An aspiring raiding Holy Priest should look closely at gear that has Intellect, Spirit, Spellpower, and Critical Strike rating. Hit rating is completely useless to a healing Priest. Haste is nice to have, but not as important since Holy Priests use a lot of instant spells. MP5 is always a good thing to have, but a robe that has Spirit and no MP5 is much better than one that has MP5 and no spirit, in the majority of cases.
Gearing a Discipline Priest
Discipline Priests love haste. Since Power Word: Shield, Flash of Light, and Penance are the main spells of a Disc Priest, and two of them have a cast time, getting them to go as quickly as possible, and have as short of a cooldown as possible, is key. Critical strike chance is also important, but haste should be favored. Discipline Priests gain no extra bonus from Spirit the way Holy Priests do, so will often opt for items that have MP5 instead. A mod, such as Rating Buster, can help you determine whether the Spirit on an item will grant more MP5 than an item with MP5 and no Spirit. Again, hit rating serves no purpose.
Gearing a Shadow Priest
Shadow Priests get a lot of benefit from Spirit and Critical Strike rating. Like any caster, Spellpower is the stat behind their damage, but the other two listed here add much to the class. The one important thing to remember is that, as a DPS Caster, you need a lot of hit rating. In order to hit a boss level mob as much as game mechanics allow, you need 17% additional chance to hit. You can get this through talents and debuffs offered by other classes, but you must ensure that you will always have that 17% chance. For this reason, a lot of DPS-oriented cloth gear has hit rating - and you should pick it up.