You'll need a Bachelor's Degree and then you'll need to apply to a vet school like Texas A&M. In total, expect to have 8-9 years in college.
Here's the Texas A&M website for their prerequisites.: http://vetmed.tamu.edu/dvm/future/prerequisites
And here are their eligibility requirements:
As soon as you can, find a vet clinic that will let you volunteer or intern there so that you can make sure it's really what you want to do.
Courtney recommends the following next steps:
You will need good grades in high school science courses, experience with a veterinarian or animals (voluteer or employment), admission to an approved undergraduate program where you can complete the prerequsite courses, and the ability to pay via savings, loans, or grants. You can find information on applying to TAMU College of Veterinary Medicine here: http://vetmed.tamu.edu/dvm/future.
Todd recommends the following next steps: