I did a similar trip this year (2009) and thoroughly enjoyed my ride through the Badlands and Black Hills - it was one of the best parts of my trip. I grew up in South Dakota (now living in Ft. Lauderdale) and was amazed at how much more enjoyable the area was to visit on bicycle.
To answer your question - I rode the Cowboy Trial (or the nice road right next to it) to Newport, NE and then up to Burke, SD - on to Winner, SD and then to Presho, SD where I picked up the road that runs next to I-90 almost all the way to Wall, SD. I didn't go into Wall the first day but rode into the Badlands and camped there - then rode though the Badlands the next day - on to Wall, SD where I camped at a nice campground just to the S. side of Main Street. From Wall I had to ride I-90 into Rapid City (legal in South Dakota). The next day I rode to Mount Rushmore (but didn't stay there long) and continued on to Crazy Horse Monument which is right on the Mickelson Trail. I rode the Mickelson Trial (fantastic!) to almost Lead / Deadwood and then got on Hwy 14 to Spearfish, SD and all the way across Wyoming. Mount Rushmore is a steep climb but not too long and certainly beautiful.
I agree with the other posters that you continue on to Wyoming like I did unless you are looking for an easier (and possibly less exciting) ride. The Big Horn mountains were the most challenging of my trip especially with the weather conditions I encountered at the top, but also the most rewarding. Yellowstone was beautiful but very biker unfriendly with narrow roads WAY too much traffic and Huge RV's taking up the whole road - but I would possibly do it again. It was suggested to me to rather take a road north of Cody, WY which is a very scenic bypass of Yellowstone and I can get that information for you if you would like.
On my blog you can see a google map of my entire route. It was a fantastic trip. http://bikedenbec.blogspot.com
The google map is here:http://maps.google.com/maps/ms?ie=UTF8&hl=en&msa=0&msid=113308409966478718980.0004642c44ff67cc85ae5&ll=37.439974,-101.25&spn=28.815161,50.361328&z=4
I'd be happy to answer any other questions for you if needed.