Enjoy the ride! The gateway towns can be good stops for meeting folks coming into the park and leaving. My fav is Red Lodge, MT. (downslope ffrom Cooke City) Cody, WY next best. An extra day is worth it at those two. Jackson, WY can be la tourista grande at the heavy traffic months.
The wife and I spent several days touring Yellowstone on our bikes right after she got her license. We Loved it! Be aware of the wildlife though, Buffalos don't play!