We've stayed many times IN Yellowstone with our 40' motorhome - in the dry campgrounds such as Mammoth (our favorite), Madison and Bridge Bay. We've also stayed at Baker's Hole national forest campground just outside the west entrance.
Here's some good information on all the roads leading into the park:
http://www.rvtechmag.com/travel/4_yellowstonebyrv.phpDefinitely go into adjacent Grand Teton Nat'l Park. Both are completely different in looks and things to do. Awesome trails of all abilities. Taking the gentle float trip in the park down the Snake River and you might see a moose in the water with you. We did!
All national parks have an awesome web site with lots of good information on things to do, camping, special alerts, road repairs, etc. Explore the site thoroughly when you're planning. Here is Yellowstone's:
https://www.nps.gov/yell/planyourvisit/index.htm