Gathering professions can be learned at level 1 (Herbalism, Mining, Skinning). Crafting professions can be learned beginning at level 5. Archaeology is available at level 20. You can have any combination of Gathering or Crafting Professions (for a total of two), yet you can have all the Secondary Professions (Cooking, First Aid, Fishing, and Archaeology).
The most popular question is how to unlearn a profession once you've learned it, in case you want to switch out and try something new. The fix is quick and simple: open up your spellbook at the bottom of your screen, click on the 'professions' tab, then click on the circle next to the Profession you want to discard.
Notes: You cannot unlearn the Secondary Professions. You also will 1) start out with your new Profession at a skill level of '1' and 2) if you decide to return to your old Profession, you will start all over again at a skill level of '1'