As noted, there are few biking opportunities in Yellowstone.
Here is a link to some of the rides that are available in the park. Many of these routes are more suited to mountain or hybrid bikes. The old road to Lone Star Geyser is interesting in that you get into a backcountry thermal area with a geyser and some hot spring pools.
In the Black Hills, check out the
George Mickelson Trail. You can ride parts of it on various days. It's an old railroad grade so the ride isn't very steep anywhere. If you ride from the Mystic trailhead, you're near the four tunnels on the route -- always a sure hit with kids.
Map and
Trail Guide are well worth having to enhance your riding experience.
If you're going this year, I do hope you already have reservations as it's pretty late to be getting them now. You may want the "Wow" factor for places to stay, but you'll be getting the "what's left" campsites at this point. In Yellowstone there are some first come, first served CG's that tend to be more primitive and away from the main attractions. Also, Grand Teton NP has many first come, first served campsites, but that's a fair commute every day into Yellowstone from there.
Your biggest challenge will be the Black Hills. Sturgis Motorcycle Rally runs August 4th - 10th and you don't want to be there then (campsites will be sold out anyway). So you might try to hit the Black Hills on your way west to YNP. We like Custer SP CG's, but again, you're going to find it a challenge to find a good campsite at this late date.
