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The color! It burns!

As a Priest, sure, Tailoring is fine. You'll make your own armor. If that doesn't interest you, you could be a consumable professional: engineer or alchemy. Engineering would let you make fun toys to augment your low DPS or help you escape dire straits. Alchemy would boost your stats, give you even more ways to heal yourself, and eventually give you a consistent way to make gold (transmutes).

Cooking is a secondary skill, and thus you can have that, FA, and fishing. If you're oging to cook, you'll either need to fight many a beastie, or take up fishing. I don't think you'd get a lot of good out of each, but some people like to try everything, and it doesn't hurt your character at all.

I think the poster meant skinning or mining.

One thing to note about Tailoring; Blizzard has apparently made it so that while crafted goods will have decent stats, they will always just be a wee bit outclassed by boss drops/world drops of the same level. If you're going to be running instances a lot, or if you like to have the best equipment off the AH, then making your own equipment is actually not a good road to go down. If this is the case, take up a consumable-creating profession or a gathering profession.