High School
The path to become a veterinarian should start in high school. Best thing that you can do at this time of your life is get experience. Experience with animals in a medical setting is one of the major factors in the admissions process. When I say get experience I mean a ton. The average number of hours is around 2,500 for most vet schools. To better your chances of getting in, starting in high school will help build a foundation for what you will add to in college.
Some ideas:
1. Local Animal Shelters
2. Organizations (like ASPCA)
3. Wildlife Rehabiliation Programs (I worked at Project Wildlife in San Diego,CA)
4. Job at an Animal Hospital
5. Work on a Ranch with large animals*
*I might add that this is a huge thing you can do that would help your application. Vet schools want people with a wide variety of exposure to veterinary medicine. There is a high demand for food industry veterinarians so experience with cows, chickens, goats, pigs, etc... will look great on your application.
I think the cow knows whats coming next.
It also goes without saying that you need to have good grades to get into vet school. But concentrate on your high school grades and where you would like to go. Most every school has a Biology major with all of the prerequisites that you would need to get into a 4year veterinary school. So if you are a high schooler concentrate on getting experience, keeping grades up, but most importantly have fun with what you are doing and be optimistic, it'll save you stress later on.