I-80 E to Rawlins, WY, then U.S. 287 to WY 220. At Casper, you can go N on I-25 to exit 210, then WY 259 to WY 357 to WY 59 south. After about 2 miles, turn Left of WY 450 to U.S. 16. At Custer, SD turn onto U.S. 385 to SD 244. Between Custer and SD 244 is Chief Crazy Horse monument, it's got a great Native American Museum and bigger than Mt. Rushmore.
Alternate route, at Casper go South on I-25 to Exit 126 (Orin, WY) follow U.S. 18/20 to Lusk, WY, then continue on U.S. 18 to SD 89. At Pringle, SD it joins U.S. 385 which joins to U.S. 16 just before Custer.
The first route is a few miles shorter than the second, both routes are good, not mountain passes or long downgrades.
Either route is almost 10 hours driving from SLC.