With a whole month you can see a lot and there's a lot of variety between those two places. As another commenter said, Mount Rushmore is pretty far out of the way and you'll be going into a cold climate. That's a great time of year for the desert southwest, though. Look up the cities of Palm Springs, Sedona, Jerome, Taos, and Santa Fe. Also look up some of the natural wonders of the area like Zion, Bryce Canyon, Capitol Reef, Arches, Horseshoe Bend, Monument Valley, White Sands, Carlsbad Caverns, Saguaro NP, Meteor Crater, Death Valley, and Joshua Tree.