I've done the Denver/Vegas stretch and beyond about 8 times.
I agree with going I-80 to I-76 instead of I-70 to get to Denver. There are some CDOT rest stops off of I-76 in Julesberg and Sterling that are RV friendly. They both have dump stations and water if you need it. Julesberg was a bit easier to access, but Sterling has a nicer staffed visitor's center. If you do decide to stay on I-80 to go up to Cheyenne then down I-25 (not sure I'd do this, but in case), the Cabela's Headquarters is in Sydney, and they have a campground there.
There's not a ton of RV friendly places to stop along I-70 through the mountains. There is a gas station just past the Outlet Mall on the north side of I-70 we stop at in Silverthorne that's pretty easy in-out. They have diesel there, not sure if you need that. Once you get past Vail there's more places that are easier to stop at, including some more nice rest stops ran by CDOT. You can get those locations on their website. We like the one in Rifle as it has free water and a dump station and is easy in-out and has some nice trees etc., good place for a picnic lunch. I can't think of many places through here where you can stop and sightsee off the highway without a car.. It's a pretty drive even by staying on I-70 though. There is a nice state campground right off of I-70 west of Palisade called James M. Robb, Island Acres. We've stayed there before and enjoyed it.
Once you get into Utah there's some gorgeous rest stops right of the highway between Green River and Salina. Nice picture taking opportunities. Some have bathrooms as well. Both the San Rafael Swell Area and the Black Dragon Canyon View Area are my favorites.
As PastorCharlie mentioned, there's some spots with long stretches between services. If you stop in Green River, UT, don't stop at the first gas station you get to off the highway, go into the town more and it'll be cheaper. There is a Pilot Travel Center that is easy in/out and the restaurant next to it is pretty good for diner food.
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