Well this may sound funny but if you live in the country find farms that are lambing or calving. Sheep are especially easy to handle and its a great learning experience without all the danger of a large animal + they have multiple births and this will give you a great experience.
Try doing a vet tech program and get a job interning with a vet. My vet has three techs working for her right now and they're always busy!
Shadow your barn's vet, get a lot of general experience working with horses or go to your local equine vet clinic
Maybe try shadowing your horse's/barn's vet