Actually, I don't think you even need a bachelor's degree (although it's preferred) to be a vet. At Iowa State (one of the countries top programs) you just need pre-vet classes (I think it's about 1 1/2 -2 years of general classes like anatomy, biology, chemistry, etc.) before you enter the veterinary medicine program, which is another 2 years or so and you can be a DVM. I'm not sure about becoming a trainer, but I'm sure there are classes for that. I've never seen classes for becoming a dog trainer at a university before, but maybe there might be something with that at community college. I know you can become a groomer through a lot of them!
I think those would be great careers! I graduated from college last year and I somewhat regret not going into a career with animals. I'm applying to graduate school now, but if I don't get in, I decided I'm going back to become a vet (I just need to get over blood). Good luck whatever you decided to do.