Do you want the fastest route or can you take your time and go the scenic route? The route above goes through some pretty desolate, not particularly scenic areas. Personally, I would take, and have driven, the scenic route, all on paved, two-lane roads that can handle any type of RV. That would be WY 487 south from Casper through the Shirley Basin, once one of the premier WY uranium mining districts. Then at Medicine Bow, the location for the famous novel The Virginian, take US 30 west to WY 130 at Wolcott, WY. Then south through the North Platte valley to the Encampment-Riverside twin towns, with an interesting history museum and the Lazy Acres RV Park that backs up to the river (we have stayed there twice). Then WY 230/CO 125 south through secluded North Park to Walden, CO, where you turn west on CO 14, which takes you over Muddy Pass to US 40. For camping sites along US 40, see my blog
HERE. US 40 over Rabbit Ears Pass to Steamboat Springs, then backtrack on US 40 to Kremmling and CO 9 to the Green Moutain Reservoir campgrounds and Dillon.
An alternative scenic route to southbound US 40 is CO 131 from Steamboat to Wolcott, CO, then I-70 over Vail Pass to Frisco and Dillon. Stagecoach State Park is a great fishing lake on CO 131 and, after all, if you've been to Wolcott, WY, you might as well visit Wolcott, CO too! But you don't have to go all the way to Wolcott because you can turn east off CO 131 at Toponas onto CO 134. That road takes you over Gore Pass to Kremmling, then onto CO 9 to Dillon.