If you only have one trip to make to the Black Hills, I'd recommend skipping Bike Week (unless that is the experience for which you are looking). It is crazy during Surgis Bike Week, too many people and too much being charged for rooms (even if you can find one).
As for staying in Rapid City, that is a CITY. I'd recommend staying in Keystone (near Mt. Rushmore), Custer, or Deadwood. These towns are actually in the mountains (Rapid City is on the edge) and much more what you come here for. My favorite these days is Keystone.
You'll want to ride Custer State Park (wildlife loop), Iron Mountain Road, Needles Highway (not a highway), and Spearfish Canyon -- lots of great riding. Visit Mt. Rushmore, Crazy Horse Memorial (being built), Deadwood for sure, Sturgis just to see what it is like NOT during the rally -- be sure to go out to Devil's Tower also. Great rides out, both from Belle Fourche and also from Sundance.