Kearney is a nice small city. I used to travel there on business about 10 times a year. Don't know a great deal about hunting around Kearney, except mostly done on the Platte River that runs right outside of town. They get a lot of waterfowl traffic. There's a Cabelas retail&warehouse on the east edge of town, they would have more info for you and they have good tent sales.

The best part is you will be a Nebraska resident and can drive 3-4 hours and be into some western hunting for mulies and whitetail. Also residents have a shot at Bighorn sheep, Antelope, and Elk, although it is a tough draw. Nebraska has a good variety of terrain from the eastern river bottoms to the western grasslands.
Niobrara is way up north on the Missouri River, quite a ways from Kearney, but not too far from where I live. We do business with the Ponca Tribe, good people.
Not saying these days you can let your guard down anywhere, but I would say Kearney and the surronding communities are about as close as it gets.