Geneva State Park (click on link below for map) has a beautiful paved trail (it had just opened when we were there in 2009) that runs from the campground and along Lake Erie to the Geneva Resort Lodge. From the lodge, you can continue biking on the main street into the village of Geneva-On-The-Lake where there are shops, eateries, winery outlets, amusements, etc. aplenty.
The roads in the campground area are flat and good for children to bike on.
Map of Geneva State Park, Ohio