Since I just looked closer at your route, I noticed that a couple of places we really enjoyed are on the route between Lincoln and Salt Lake City. At Sidney, NE, you might want to consider camping at the Cabela's store. The store is great, has a nice small restaurant, and a very nice attached full hookup campground. If you're too lazy to walk over to the store, they'll run you over in a golf cart. You might find some goodies at the store you've just gotta have, too. A little north of the route in Gering,NE, is the Robidoux Campground, which is owned and run by the city of Gering. It is spacious, nicely landscaped, and really well maintained. Oh, by the way, it has glorius views of Scotts Bluff National Monument from your campsite and Chimney Rock is just down the road. There's also a couple of neat museums in the vicinity.
Jerry