I have been to Bryce in the past, and decided to go a brief road bike ride through the park on my way from Zion to Moab. I road from the entrance parking lot to the Far View Overlook. There are not a lot of views into the canyon from the road so you will want to pull into the overlooks if you really want to see into the canyon. The road is well maintained, and there is a bike path that will take you several miles into the park. However, the bike path is very curvy, likely to control speed on a path designed for both pedestrian and bike traffic. On my return I stayed on the road and avoided the bike path. The shoulders were not very wide in this section, but for the most part I was going downhill so there was not as much of a speed differential between me an traffic.