Well, MT Rushmore, the Badlands and Crazy Horse are good choices, and so is Yellowstone, you will pass right by it on I90 before you take I15 N if you are going by the Interstate system.
Suggest you get RV Parky or Allstays and IExit apps for you phone/tablet. I don't normally make any reservation until a few hours out because I like to maintain maximum flexibility but June is likely a busy time in those places.
Good luck and have a wonderful trip, Have been along that route several times, It does get a bit busy around by Gary IN and south of Chicago as you likely exoerienced so I recommend not going through during rush hours. The other option is to go over the top of L Michigan. Been that way also but there are some 2 lane roads that way, but a nice drive and nice country.