Yes, But Ten Days is Really Only Six
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You can have a wonderful week seeing everything there is in the Rapid City area and make the drive down to Flagstaff to visit the Grand Canyon. Just realize that in order to do that, you'll spend a day flying west, two days driving between your two destinations, and a day flying home so that if you budget ten days for the trip, you'll really only have six days on which you can actually 'stop and smell the roses'. And even that assumes that you'll book more expensive open-jaw airline tickets along with what could be a very expensive one-way car rental. So to be honest, I'd probably do this as a 'local' RoadTrip out of Rapid City.
Among the many natural wonders in the more-or-less immediate vicinity of Rapid City, some of the best are Mount Rushmore. Wind Cave, Jewel Cave, Devils Tower and the Crazy Horse Memorial. A bit farther east there are plenty of parks along the Missouri River in South Dakota including Fort Pierre Chouteau National Historic Landmark, LaFramboise Island Nature Area and Farm Island Recreation Area.
AZBuck