Once you hit level 40, you will be able to receive a quest that will allow you to get your first aid skill above 225.

1) Pick up the quest from the First Aid Trainer in Ironforge.
2) She will send you Theramore to visit Doctor Gustaf VanHowzen. Doctor Gustaf is located at 67,48.

3) Gustaf will give you a Triage bandage with 20 uses. Make sure you know where the bandage will fall in your inventory so you can quickly drag it to the quickbar.
4) Drag the Triage Bandage to the quickbar and stand in the middle of the room. Zoom your camera almost all the way out so you can see as many of the beds as possible. 
5) Target critically injured 1st, badly injured 2nd and then the injured soldiers. Critically injured will die very quickly, so it's imperative you heal them first. Click on a soldier and then hit the hotkey for your Triage Bandage to heal them. After the healing process is started on 1 patient, go ahead and target your next patient but do not move or your bandage will be cancelled.
6) Once the Healing process is complete, hit the hotkey and start to bandage the next soldier.
7) Repeat step 5 and 6 as needed.
8) If you fail, make sure you delete the Triage Bandage from your inventory. Gustaf will give you a brand new one when you accept the quest again. 

How to get: Heavy Mageweave Bandage
Get your skill in First Aid to 240 then go back and speak with Doctor Gustaf VanHowzen, the same person that gave you the Triage quest.  He gives you the training free.

How to get: Runecloth Bandage
Get your skill in First Aid to 260 then go back and speak with Doctor Gustaf VanHowzen, the same person that gave you the Heavy Mageweave Bandages.  He gives you the training free.
 
How to get: Heavy Runecloth Bandages
Get your skill in First Aid to 290 then go back and speak with Doctor Gustaf VanHowzen, the same person that gave you the Runecloth Bandages.  He gives you the training free.