Odds of Getting In
Brutal. At least to the best schools. Veterinary medicine is one of the most hotly-contested professions in the world. Everyone wants to save Rover. But the field is growing, or at least the pets per capita is. And while it's a popular path, there aren't all that many accredited veterinary colleges, so the pool of potential vets is somewhat limited (hence the difficulty of getting in).
You may have to prove that you love animals more than other hopefuls by kissing a German shepherd on the mouth. (Theirs likely has fewer germs than yours, by the way.)