The Yellowstone area is definitely one of my favorite trips. As others have said Jackson Hole, West Yellowstone, and Cody are all good places to stay. Cooke City and Red Lodge are also good depending on how you want to break up the ride. The last trip we went north out of the park and over Beartooth, spending the night in Red Lodge. The next morning went east to Belfry, south to Chief Joseph, and back over to Cooke City. If you were camping I would say plan on spending a night at Beartooth Lake Campground (it is probably my favorite place to camp)

If you want to venture a bit east of the park 14 over the Big Horns to Sheridan is a great road and 16 coming back from Buffalo is also very good.

 

the wife and I were there in Sept. 2016, last time was back in 89. the traffic is heavier than it was years back and you have the idiots that will stop right in the middle of the road to look at something, but was still enjoyable trip. we stayed in West Yellowstone also and when leaving, rode down thru Grand Teton and stayed 1 night in Jackson, always fun to visit.

 

+1 on that. I ate there on a recommendation while I was getting work done on the bike at the local HD shop (Belgrade about 15 min or so away).