It helps to love animals and have a calm, reassuring presence around them.
Volunteering at local animal shelters or veterinary offices provides good experience and can get you "in the door" for referrals and entry-level employment.
Work as an animal trainer is also good experience if you later decide to go to veterinary school.
Most animals who compete have both a trainer and a groomer.
For information about local educational programs in animal training, check with veterinary offices, your county's extension service office, the Humane Society, and zoos or marine exhibits.