How do I get a Profession?
You have to find an NPC that can teach you their Profession. Ask other people where you can find certain trainers, or discover them on your own.

Using Professions
After you have trained or purchased a Profession, hit "P" to bring up your abilities book. Then, look for a new icon that has been placed in there for that skill. You can drag this icon over to your action bar to make it easier to find and use. If you can't fit it on the first row of your action bar, place it on a different row by left-clicking on the scroll arrows to the right of your action bar (it's ok if the icon is still selected on your cursor).

Profession Limit
You can learn all of the secondary skills, but can only choose two professions.

Character Level Limits
In order to prevent low level characters from mastering high level skills, a certain character level limit is required for each level of profession. You will need a minimum character level to learn a new level of professions. Apprentice requires level 5, Journeyman level 10, Expert level 20 and Artisan requires level 35.

Professions If you go Mining there are two recommended paths. Decide which one you want to take.

Mining - Mining is used to collect items for engineering and blacksmithing.
Blacksmithing - Create armor and weapons.

or

Mining - Mining is used to collect items for engineering and blacksmithing.
Engineering - Engineering is used to assemble cloth, leather, metal, and stones into parts needed to make explosives, guns, scopes, bullets, mechanical dragons, aquatic helmets, and more.

Herbalism/Herb Gathering - Collect herbs that can be used with alchemy.
Alchemy - The alchemist mixes herbs to generate potions with a variety of effects.

Skinning - Gather leather and hides for leatherworking, blacksmithing and tailoring (mostly leatherworking).
Leatherworking - Create leather armor and armor patches.