Why go thru Casper? Take I-25 N. to exit 126 in Wyoming, then take 18-20 toward Lusk, Wy, then 18-85 N to 18 turn off, go thru Hot Springs then take 79 N. into Rapid City. This keeps you off the tourist route and across good federal and state highways - entering Rapid from the south.
Make fuel fill stop in Cheyenne, and you can avoid the higher prices in Lusk, WY and Hot Springs, SD - follow hwy 18 truck by pass at Hot Springs to avoid the traffic lights in old town.
Note that you will encounter 85 octane reg unleaded in states above 5,000 ft. elevation. Lower oxygen % at this elevation means "thinner" air, which lowers effective compression ratio in engine and therefore the 85 octane works fine. (85 octane not to be confused with the E-85 fuel which is not widely available anyway. )
Lots of tourists in Rapid this time of the year too; watch out for unexpected u-turns and wrong way drivers on one way streets in downtown. If you want to get away from the steel and glass modern hotels, go downtown to the old Hotel Alex Johnson - built in the 1920's, still authentic and rustic, great wine and micro beer bar and dining room. Lots of free parking on the streets, just watch the two hour time limit - they even have "meter maids" in the old west!!! lol
Rapid Chevrolet local dealer excellent service department, on Deadwood exit I-90 west of central Rapid City; very accommodating to "out of towners" on a tight schedule.