Yellowstone was fun when we passed through while doing the Trans America. We liked it a lot. There lots of good sights to see within a short hike from the road, or even from the road itself. There is a lot of cool looking geothermal stuff of various descriptions, also beautiful waterfalls, etc. Wildlife is plentiful.
The roads are OK. They are fairly narrow and mostly without shoulders, but we didn't find that to be a huge problem. It might have been worse if we had passed through on a weekend.
It isn't completely clear if you are bike touring or car camping with bikes along. If bike touring, campsites are guaranteed even if the campgrounds are otherwise full.
BTW: The Tetons are close enough to tour both Yellowstone and the Tetons in the same trip if you have a little time.