I'm going to read your inquiry a bit differently, that you're saying you're wanting to get there as quickly as possible, and then spend time in the area. (Otherwise, as others have said, that's a LOT of driving and little time to see sights).
Once you get there, the actual visit to Mt. Rushmore can be done in half a day. There's the usual photo ops on the plaza, and there's a hike around the base of the monument. There's a theatre that shows a film on the makings of the monument, it's history. There's other historical information on site as well. But still, a good half day and you're done. But also plan on seeing it at night as well, when it's all lit up.
In the area: Crazy Horse Monument. Custer State Park. Jewel Cave National Monument. Wind Cave National Park. Needles Highway. Devil's Tower National Monument. And those are the ones that immediately come to mind.
H E R E - Black Hills/Badlands is a link to many things to see and do in and around the Black Hills region.