You may only have 2 of the main production/gathering skills at a time. That means that you must choose two from: Balcksmithing, Leatherworking, Alchemy, Tailoring, Enchanting, Engineering, Mining, Skinning, and Herbalism. You may choose to unlearn one of them to replace it with another at any time. Your new skill will begin again at level one. If you ever choose to return to the skill you dropped in the first place, it will begin again at level one as well. If you unlearn a skill you will also lose all the recipes that you bought and trained for it, even if you return to it later, you will have to relearn all of them.
In addition to choosing two of the skills from that list you may also do all of the following: Fishing, Cooking, and First Aid. These three skill do not count towards any slots and any character may learn all of them at the same time while still having points in their two main skills.