Primatologists - work with primates; Herpetologists - work with reptiles; zoologists - all animals; etc. There's not a single name for someone that works with exotic animals other than the broad category of zoologist. I'm a zookeeper and work with all kinds of exotic animals. I did an internship at a Vet Clinic but was bored because it was always cats and dogs and same thing every day. If you want to be a vet for exotics, you might consider being an in-house vet at a zoo. You can do internships and volunteer work at zoos to get exposure to the exotic animals which will help you decide which direction you want to go. As a zookeeper, I get to take care of; train; and provide basic medical care for hyenas, porcupines, capybaras, baboons, spider monkeys, cappuchins, lemurs, wolves, bears, bearcats, sloths, tamanduas, etc. I love it!
Courtney recommends the following next steps:
There are a variety of people that work with exotic animals. If you are interested in the care of animals, you may want to consider a career as a zookeeper or aquarist; animal trainers and marine mammal trainers care for animals and study the behavior of them. Park rangers and guides work all over the world, including on the ocean, protecting and teaching people about animals.
It is important to decide what type of animals you are interested in and then study the behavior and the ecosystems for those animals. Experience in volunteering at zoos and aquariums, internships in research, and study abroad opportunities will help with gaining experience and helping you focus on what animals you are interested in.