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I have family in Sheridan! The last time I went through, I took the Yellowstone route, and spent a day there and the night in Bozeman, MT. If you're pressed for time, I'd take 87 up to I90, but Yellowstone is worth the delay if time isn't of the essence.
Honestly, from Sheridan to Portland you don't have a good direct route, and I'd take I90/385 through Spokane then I84 to Portland (Google agrees, fwiw.) The only reason I say stick to the interstates is that I recall some of the small roads having no problems turning into wonderful cliffside views.
photo by splunge
photo by TheophileEscargot
photo by Kronos_to_Earth
photo by ethylene
