I would take I-25 south to Pueblo, US50 East to Dodge City, KS, then East US54 to Nevada, MO, US71 South to Joplin, MO, then I-44 East to Springfield, MO and then US65 South to Branson.
Sites along the way: Bent's Old Fort at La Junta, CO, Dodge City has a Western thing downtown, Prescious Moments Chapel near Carthage, Mo, Bass Pro in Springfield. There are also a couple of Civil War battlefield around Carthage, MO and Springfiled, MO.
I would avoid camping in Garden City, KS. The one I recalll is close to a feedlot. Gunsmoke CG is at Dodge City. It is pretty good. East of there it is kind of limited. It all depends on what you are looking for.