Hwy 16 over the Bighorns is relatively easy. It's a 4-season, wide, 2-lane highway with few sharp curves and no really steep climbs or descents. The mountain driving is between Ten Sleep and Buffalo (about 60 miles). It's a pretty drive with lots of USFS campgrounds (no hookups), or if you need hookups there are commercial campgrounds in Buffalo (and I believe also in Ten Sleep).
I'd highly recommend camping in the Black Hills over the Badlands. One of our favorite spots is Devils Tower, WY, the nation's first national monument. We prefer the campgrounds within the "park's" borders, Belle Fourche River Campground, but if you need hookups, there's a KOA just outside the gate that's also nice.