Qualifications
This is one profession where experience is what gets you noticed, and while there are schools that will certify you in dog (and other animal) training, you don't necessarily need a college degree to be an animal trainer. That being said, while a college degree isn't required, it's preferred when it comes to certain types of jobs for specific types of employers.
If you have your heart set on working with, for example, aquatic animals at Sea World or Busch Gardens, where hundreds of people apply for ten or fewer open spots (source), a degree in animal husbandry, animal science, zoology, or a related field is what will get you noticed. Many professional companies expect that you will have completed, at the very least, some coursework in biology, marine biology, zoology, and animal behavior or psychology. And don't forget getting your SCUBA and CPR certifications, as well as basic first aid skills.
As useful/necessary as all of that is, though, the majority of the training you get will come from hands-on experience. You'll learn from someone more experienced than you and then take that knowledge with you. Other trainers are your most important resource.