I live in NW Nebraska. In general we have Pheasant, Coyotes, Skunk, Badger, Pronghorn, Whitetail and Mule deer, Prairie Dogs, Grouse, dove, fox, elk, and bighorn sheep. I'm sure I missed some too. Elk is a once in a lifetime draw. Meaning you get an Elk license you can't ever get another one. Bighorn sheep were reintroduced about 10 years ago. I think they have 2 licenses a year. 1 they draw and the other is auctioned off.
Go to
http://www.ngpc.state.ne.us/hunting/hunting.asp to get good details. There is quite a bit of public land in Nebraska, but of course the best hunting is on private.
I went out Prairie Dog hunting a couple of weeks ago and saw Pheasant, Coyotes, Pronghorn, mule deer, whitetail and of course Dove and Prairie Dogs. Land owners are getting a lot more stingy with hunting rights. If you work for a grain company or some other agricultural company where you get to know a lot of landowners your chances of getting permission will go way up.