First Aid is probably the easiest profession to level to 375, you can probably do it in one sitting. It doesn't require a secondary skill and anyone can farm the cloth drops themselves but you can also buy them off the Auction House.
There will be several manuals you will have to purchase and a quest you will have to do while leveling to 375.
Lots of Linen cloth, if there isn't any for sale on the auction house take a high level character and clear out some humanoids in the 10-20 range.
You'll start using Wool cloth in this step. These are dropped off approximately level 15-25 Humanoids if there isn't any for sale in the Auction House.
To go from 150-225 you will have to purchase Expert First Aid Manul. If you are Alliance you can get it from Deneb Walker in Stromgarde Keep in Arathi Highlands. For Horde, Balai Lok'Wein in Brackenwall Village in Dustwallow Marsh has it.
To get your skill from 225-300 you will have to complete a quest. The pre-quest is available from the Horde Trainer in Orgimarr and the Alliance Trainer in Ironforge. The Horde quest continues to Doctor Gregory Victor in Hammerfall of Arathi Highlands. The Alliance quest continues to Doctor Gustaf VanHowzen in the Foothold Citadel in Theramore of Dustwallow Marsh.
There is another manual that you have to purchase to go from 300-375. If you are horde it is located at Falcon Watch in Hellfire Peninsula from Aresella. If you are Alliance it is located at the Temple of Telhamat in Hellfire Peninsula from Burko.