I've had a lot of questions about Professions from newcomers and players who are in the higher levels aswell and it seems to be a good idea to have guide here to for those who need awnsers. I'm copy pasting most of this info from the official WoW webpage.
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. The easiest way is to ask guards in major cities.
How do Professions work?
There are two types of skills in World of Warcraft: gathering skills, which allow you to collect the raw materials needed to make items, and trade skills, which actually transform those materials into finished goods. Trade skills are generally paired with gathering skills in a relationship that makes each integral to the other. To make items with the blacksmithing ability, for example, players will need ingredients that are acquired by applying the mining skill. Players will not have to be equipped with both paired abilities to succeed, however. An alchemist working to create a potion can always purchase the necessary herbs from an herbalist.
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).
How do players acquire new recipes?
Trade skill recipes are the blueprints for crafting an item, and they detail the raw components required to manufacture that item. Many recipes are available from the same in-game trainers who teach a specific Profession. Other recipes are found through various vendors around the world or on the corpses of slain monsters.
Since you can take 2 professions there are a few recommended paths to follow:
Mining - Mining is used to collect items for engineering, blacksmithing and jewelcrafting.
Blacksmithing - Create armor and weapons.
Mining - Mining is used to collect items for engineering, blacksmithing and jewelcrafting.
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.
Mining - Mining is used to collect items for engineering, blacksmithing and jewelcrafting.
Jewelcrafting - Create rings, necklaces and trinkets.
Herbalism - Collect herbs that can be used with alchemy.
Alchemy - The alchemist mixes herbs to generate potions with a variety of effects.
Herbalism - Collect herbs that can be used with alchemy.
Inscription - A scribe mills herbs into ink to create scrolls, books and glyphs.
Skinning - Gather leather and hides for leatherworking, blacksmithing and tailoring (mostly leatherworking).
Leatherworking - Create leather armor and armor patches.
Tailoring - Create cloth items and bags.
Enchanting - Improve existing items.
Secondary Skills
All secondary professions can be trained.
First Aid - Healing through bandages made from cloth.
Fishing - Collect food. Good for cooking.
Cooking - Food is used in the game to heal players out of combat so that they can return to the fighting as quickly as possible, some cooking recipes give bonuses.
Skill Increases
Gray - You cannot increase your skill by making these items or harvesting these items.
Green - These items give you a fairly poor chance to increase your skill. You can still do it, but its going to take a lot of gathering/creation to up your skill with these items.
Yellow - These items give you a reduced chance of increasing your skill. It's still pretty reasonable to try to increase your skills with these items.
Orange - These items give you your best chance of increasing your skill. For many of the Professions, making an orange item will always increase your skill.
Red - Your skill is not high enough.
Secondary Professions Training
Expert and above skill tiers for secondary skills require you to find the books to teach you them. Artisan level requires a quest to be completed.
Tips
Some Professions allow you to use the trade screen to interact with other people's items. For example, enchanters can enchant items directly in the trade screen. Lockpicking and some other Professions can also be used via the trade screen.
Know which classes will need the products that you can create with your Professions:
Mail: Warriors, Paladins (Hunters and Shaman can start wearing mail at level 40)
Leather: Rogues, Hunters, Druids, Shamans
Cloth: Mages, Priests, Warlocks
Know which Profession components to save while you are fighting. You can sell them to other players or give them to friends, rather than selling them to the merchants:
Blacksmithing: jewels, metal bars, linen, wool, leather...
Engineering: jewels, cloth, leather, mechanical items, metal bars...
Leatherworking: leather, ruined pelts, hides...
Tailoring: linen, cloth, wool, silk...