Have a wonderful trip! Some good suggestions from others. Cloud Driver is right, US 26 across eastern Wyoming/western Nebraska has some treats you won't find on I-80. Very nice county-run CG near Scotts Bluff too (Robidoux RV Park), with hookups. It's popular; you may need reservations there if you're interested.
US 16 definitely the best route over the Bighorns. If you do decide to stay on I-90, be sure to make some time if possible for either Devils Tower (off I-90 perhaps 35 miles or so) or Little Bighorn. Well worth it. It's the longer way around from the Black Hills to Cody, but probably not THAT much slower, as it's not mountain driving.
Enjoy!