As soon as your character hits level 5, it's time to pick up a profession -- or two or three! Each character you have can take two primary professions -- herbalism, mining, skinning, alchemy, blacksmithing, enchanting, engineering, leatherworking, tailoring, jewelcrafting, or inscription -- as well as however many secondary professions -- cooking, fishing, first aid, and archaeology -- they want. Primary professions are typically categorized as "gathering" professions -- herbalism, mining, and skinning -- that allow you to collect materials and "crafting" professions -- alchemy, blacksmithing, enchanting, engineering, leather working, tailoring, jewel crafting, and inscription -- that allow you to create items.
When choosing a primary profession, it's good to pick two that work well together: say, a crafting skill that uses the items you collect with a gathering skill. This means herbalism is typically paired with alchemy or inscription; mining is typically paired with blacksmithing, jewelcrafting, or engineering; skinning is usually paired with leatherworking; and enchanting and tailoring can be paired with anything (though often players will pair them together and use goods created by tailoring to level their enchanting). But if you aren't terribly interested in crafting or just want to give your character a leg up with money-making, you could pick up a pair of gathering professions and take whatever you gather while you're leveling to sell to crafters on the auction house. As to secondary professions, since you're not limited on the number you can take, you can really grab as many as interest you.
Not sure where to start with all these choices? We'll walk you through it.