With your Class B, a do-not-miss drive in Yellowstone would be US 212 from Red Lodge to Cooke City, MT. It is called the Beartooth Highway and is one of the most spectacular drives in the U.S.
If, as several people have suggested, you do take US 191 south from Jackson, WY, a great campground is the USFS Granite Creek Campground a few miles east of the highway and south of Hoback Junction. It is on a long gravel road, but has some spectacular views the surrounding mountains. We had a campsite with a water hookup, but there are no other hookups and no dump. However, this cg may be too close to Jackson for your trip.
We also stayed at the USFS Firemen's Memorial CG on US 191 in the Uinta Mountains of Utah. Nice cg, but the dump fee is not included in the cg fee (it is operated by a concessionaire). But there are some spectacular views of Flaming Gorge Reservoir from the highway. And the Utah Fieldhouse Museum in Vernal is definitely worth a several hour stop.
