before retiring and escaping California, I used to do that route 3 times a year to my 2nd home in the Black Hills.
I-15 to LV & SLC, then 80 E is the right general route and almost all interstate driving. the 2 main options are how you get through SLC and how you jump from eastern WY over to SD.
coming up to SLC from the south, I always preferred to get off I-15 before SLC and use the Provo Canyon Hwy 189 up to Heber, past Park City to hook up with I-80. It avoids the SLC traffic jams, it avoids the steep climb up Parleys Summit on I-80 (although 189 is also a climb, but not quite as steep), and most of all it gives you a scenic break from the long boring hours on the interstate.
There are several routes to get from Wy over to SD and I used them all, depending on whether I was headed to Rapid City, Sturgis or my cabin in the hills. Usually I would take 287/220 after Rawlins, up to Casper then get on 18/20 over to Hot Springs SD and north to Custer or Rapid City.
