I'm nearly finished gearing up my newly minted priest for raids. Here's how I've been doing it.
With the new heroics and Looking For Raid, gearing up isn't really that hard or complicated now. It does require a burst of activity though, and you'll quickly get sick of Vortex Pinnacle.
The first thing you want to do is get your hands on the highest ilvl gear you can wear. That includes crafted pvp gear - it may have icky resilience, but the intellect means it's far superior to anything you're wearing. Ask in guild and check the AH for anything you can to boost your ilvl. Then grind dungeons, heroics, and Hour of Twilight Heroics in that order to get your maximum valor points per week. Use the valor points to replace whatever the lowest level gear you have is - most likely, a trinket.
Use the justice points to pick up ilvl 378 tier gear. And once your gear is of a high enough ilvl, start queuing for Looking For Raid. Only at that point is it worth considering stat weights and what gear is actually better for you. Before that, it'll only slow you down and not make a huge difference anyway.
Don't be afraid to use spirit gear either, if it's a) significantly better in terms of intellect than what you already have and b) you're not nicking it off someone who actually uses spirit.