I volunteer at our zoo here, and I get asked this question all the time (well, not for Disney, but how to be a zoo keeper...)
Go to college....graduate! :lol: Degrees is Animal Science, Biology, etc are very helpful. Get experience with animals.....volunteer at a zoo, an animal shelter, animal rehab center etc. Many zoos and other facilities have internship programs, or even summer seasonal positions, look into these. There are also some intensive exotic animal colleges like Morpark in California....good to look into after you have your BS.
Strike up conversations with zoo keepers in your area, they will also be a wealth of information. It's never to early to start....good luck!! :sohappy: