As you prefer 4 lane highways, what Jerry advised would be what I would also suggest. We go to Rapid City and the Black Hills often and have used that route either coming or going several times.
I would like to make you aware that the Sturgis motorcycle rally is held this year the first week or so of August. The 3rd to 9th I believe. It is always a big event but this year they will be celebrating the 75th anniversary and the area will be more crowded than usual. Without reservations, it will be difficult to find a campground to stay at.