Personally, I would go to Mohican State Park. It's probably an hour from Medina off I-71, and it offers everything you're looking for and much more.
The Class A camp area has above-average amenities including an outdoor Olympic-sized swimming pool. If you want to go primitive in Class B, you can pay a fee to use the Class A facilities.
http://www.mohicanstatepark.org/camping.htm
http://www.mohicanstatepark.org/park_activities.htm
Mohican State Park has both a mountain bike and regular bike trail.
http://www.dnr.state.oh.us/parks/parks/mohican/tabid/769/Default.aspx
Horseback riding is available in the area and bridle trails are available in the park.
http://www.loudonville-mohican.com/activities-rec-horseback.html
Other public campgrounds are available at nearby Pleasant Hill Lake Park and even a few primitive campgrounds at the also nearby Malabar Farm State Park. Additionally, there are very acceptable private campgrounds in the Loudonville area, but they don't offer all of the same amenities or convenience as Mohican State Park campgrounds.
http://www.mwcd.org/places/parks/pleasant-hill-lake-park
http://www.dnr.state.oh.us/parks/parks/malabar/tabid/762/Default.aspx
http://www.loudonville-mohican.com/accomm-campground.html
Note that at Pleasant Hill Marina you can rent power boats and pontoons for fishing.
http://www.pleasanthillmarina.com/
The Mohican River area is beloved for its family canoeing. Unless there are heavy rains, the Mohican and its tributaries are laid back and adults can stand easily in many locations.
The Mohican River and Pleasant Hill Lake both offer fishing opportunities.
If you want non-camping activities, Malabar Farm is both a working farm and a piece of Hollywood history frozen in time. Other activities are available, including occasional barn dances.
Loudonville is a tourist attraction and a gateway to Ohio Amish country. Nearby Mansfield offers the Mansfield Reformatory, famous as a movie set, for its architecture and for its alleged ghosts and definite spookiness. Also at Mansfield is the nation's only new hand-carved carousel opened since World War II.
WARNING: State park campground reservations go quickly!
BOL for a great and memorable family camping outing!!!