I have lived in Wyoming more than 20 years and have biked nearly every paved, non-Interstate mile of pavement in the state. I would urge you not to ride from Cheyenne to Rawlins, but rather to ride from Scottsbluff through Casper. In the Panhandle, Hwy 92 generally has far less traffic than US 26. Hwy 29 in far western Neb is a lovely, empty road - then connecting with US 20 heading west. I prefer that to using US 26 thru Torrington to the service roads paralleling I-25. From Orin Jct to Douglas there is a parallel paved service road - then Hwys 93 & 95 keep you from having to ride on I-25. US 20/26 heads west to Shoshoni - great milkshakes at Yellowstone Drug - and Riverton and you can then continue on over Togwatee Pass along the TA.
Be aware that, heading west, you will encounter significant headwinds which are almost always worst in the afternoon. If you are doing a loop with a return trip, I would suggest the section thru Sylvan Pass to Cody - Wapiti Canyon along US 14/16. If you have your cycling legs by then, I might suggest continuing over the Bighorn Mountains - fabulous wildflowers - to Buffalo and then via Kaycee and Wright to Newcastle - then through the southern Black Hills to Hwy 2 and to Crawford, Nebraska.
Hwy 2 in far northwestern Neb is empty and magical.