I'll be interning at IL's wildlife clinic for the summer, and right now I'm co-chair of the clinic and volunteer there. We get animals like pigeons, squirrels, songbirds/waterfowl, rabbits, all kinds of raptors, opossums, raccoons, deer, and coyotes/mink. That's about the order of frequency that these animals come in, from most to least.
There are many wildlife rehab places that each have a vet or two on staff. Usually just one, though. You usually deal with native wildlife, and you see a lot of hit-by-car cases. Most are going to die or you'll have to euthanize them... I think the national success rate hovers around 40%, and that counts quite a few "orphaned" animals that are healthy when they get to you. You set a lot of broken bones. You deal with a lot of public education. You work with a lot of rehabbers, some good and some not so good. You probably will have research projects on the side too.
There are some really neat facilities out there. You should try volunteering at one, they almost always accept volunteers (ours are required to be rabies vaccinated first though). As for job opportunities and training, most vet schools seem to have a wildlife center affiliated with them, and you can do plenty of externships 4th year (depends on your program's flexibility with you going off-campus). Where you find a job depends on openings... you won't have your choice of cities but jobs are out there!
