Here's what you'll do to get your skill up. (You can find a list of trainer locations here.)

Beginnning characters can level their cooking skills as they go and fishing is a big help there. You catch fish and cook them up with the recipes you get from the trainer. Easy as can be.

Here's one route to follow to get your skill up. Recipes that are marked with an asterisk '*' appear in the MMO cookbook, allowing you to make them in-game and in the real world.

if you are higher level, and have the cash, you'll be able to find most of the ingredients listed below on the Auction House.

Alliance side (the Horde Guide is below:)
1-40: Cook up the starter recipe, Spice Bread*. Get the flour and mild spices from the vendor and cook till your skill hits 40.

Get the boiled clams recipe and the crab cake* recipe from a trainer, head to Darkshore, and kill all the crawlers (or any sort) along the beach. Collect the clams and the crawler meat. Cook 'em up and get your skill to 75. You'll need to buy Journeyman cooking when you get the chance, before you hit skill 75.

Alternatively, you can go fishing and catch a bunch of rainbow fin albacore and cook those.

Cook the crab meat till you hit 125. Then find a trainer and get the Goblin Deviled Clams* recipe, head to the Wetlands, and kill a ton of Bluegills. They'll drop the clams with the tangy clam meat. You'll want about a hundred pieces of the clam meat. Then cook till you hit 200.

Buy, beg, or steal the recipe Spider Sausage and head off to Dustwallow Marsh where you'll kill Darkmist Spiders (by a cave to the NW) and collect the white spider meat. Cook that till you hit 225. Then do the Artisan Cooking quests.

Once you're done with the Artisan quests, for 225 - 285 or so you'll be cooking eggs, specifically the Monster Omelet (1 x Giant Egg, 2 x Soothing Spices.) This might require a lot of eggs...

from 285 - 300 go for the Smoked Desert Dumplings (1 x Sandworm Meat, 1 x Soothing Spices) x 20. You get the recipe from a quest and a skill of 285 is required to get it.
Note: You'll need to buy the master cooking book off of Father Malgor Devidicus located within Honor Hold/Hellfire Penninsula before you can go past 300 cooking skill.

300 - 325 Ravager Dog* (1 x Ravager Flesh).

325 - 350 Recipe: Roasted Clefthoof* (1 x Clefthoof Meat) x 35-40

350 - 375 Recipe: Spicy Crawdad (1 x Furious Crawdad) x 30-40