My source, WoWhead. Presented here in more condensed form, because I got tired of scrolling.

Expert First Aid (150–225) is learned from Deneb Walker in Stromgarde Keep, northeast side by the chapel (27.2, 58.8) (Alliance), or Balai Lok'Wein at Brackenwall Village in Dustwallow Marsh, between the caravans on the north side (36.4, 30.4) (Horde). These vendors also sell the recipe books for Heavy Silk Bandage and Mageweave Bandage.

Artisan First Aid (225–300) is learned from a quest. Upon reaching level 35 and First Aid Skill 225, the optional breadcrumb quest Alliance Trauma/Horde Trauma will become available. The breadcrumb quest can be skipped, and you can go directly to Doctor Gregory Victor in Hammerfall, Arathi Highlands or Doctor Gustaf VanHowzen in Theramore, Dustwallow Marsh for the quest Triage/Triage, from which you get Artisan First Aid.

With enough Mageweave, it's possible to get to 300 with only Heavy Mageweave Bandage. (It turns green at 270, grey at 300.)

Outlands and TBC:

Created 2021-11-14, modified 2021-11-14.