Is there a bus that goes from the airport to western Nebraska, specifically Kimball or Sidney, NE?

You can also check boutiqueair, it might have flights to Sidney.

You could take the train form the airport to Union Station, and then the Amtrack to Omaha. There also is a company called Bus Express Arrow, but I doubt they go out of Denver Airport.

You'll have to look up the bus routes or call the bus company