I'd recommend Making of a Woman Vet by Sally Haddock. Came out in 1985 but still very good. She talks about her time in vet school, learning the ropes, treating patients, etc.
Another good one is If Wishes Were Horses: The Education of a Veterinarian by Gage.
Herriot's books ARE wonderful, but they're about being a vet in the 30s-60s in England. They're more about the people and animals than how to become a vet.
Good luck!