In addition to what everyone else has said, check out organizations like worldwide experience or brothers safari. I just went on their Vets Go Wild course and it was AMAZING. WWE offers other 'gap year' courses, and I think Vets and Wildlife (used to be Go Bush Direct) does too (not just for vet students).
While I was on the VGW course, a vet helping with the course that is now a LA vet, but has done some zoo work, gave me the following advice. Once you graduate, go work in a small animal clinic, and offer to take on any of their exotics patients. A lot of strictly small animal vets don't like exotics patients, so they will probably be thrilled (per him). This gets you elementary experience. Then, join an exotics practice, to beef up on your exotics medicine. You probably won't be a brilliant surgeon, or internal medicine vet, or whatever, but a good all-round practical exotics vet. People with solid exotics experience are often the ones (again, allegedly) that get called out to help treat zoo animals (I presume assuming they don't have their own vet-Metro Toronto Zoo does, but I think I remember the Dick Vet treating animals at the Edinburgh Zoo and 3 Sisters Zoo). He did NOT recommend differentiating between zoo and exotics, though that is just his opinion.
I would recommend doing a placement with a zoo or wildlife vet, as it can be quite different. Most of what I saw this summer was not what I would consider 'medicine' per se, but more preparing animals for safe transport to another facility. This might be a difference between wildlife and zoo medicine, but at a wildlife park they are supposed to live natural lives, and the vets are very hands off. Depending on the value of the animal and how many there are etc., they may not even treat if an animal gets hurt or sick. That is not to say they don't get to do really coold things, because they do!
Oh, another place to see about volunteering is a wildlife rehabilitation center. Even if you go in as just the cage cleaner/laundry/etc., you can often work your way up to caring for the animals, and I think the one I did is one of the BEST placements, both for enjoyment and learning, I've had.
Good luck!