How much are you willing to drive each day? A possible itinerary could be:
1: Drive to Chicago, can do Fort Wayne and/or Indiana Dunes National Park on the way
2: Brookfield Zoo in Chicago. A large zoo with exciting species like pangolin
3: Drive to Omaha, can do Blank Park Zoo in Des Moines or stop at Nachusa Grasslands or any number of wildlife areas along the way
4: Do Omaha's Henry Doorly Zoo. Massive zoo that stands as one of the best in the country and even world
5: Drive to Kansas City, can stop at Loess Bluffs National Wildlife Refuge, can try other things around KC if you get there early
6: KC zoo, a large zoo that is well known for its Africa section, opened a new aquarium, drive to ST louis
7: St. Louis Zoo, drive to Memphis
8: Memphis zoo, really uniquely themed zoo, can also see the Bass Pro Pyramid on the Mississippi which is cool
9: Drive to Nashville, Nashville Zoo
10: Drive back to Ohio, could do Mammoth Cave National Park in the morning
This hits: Brookfield, Omaha, Kansas City, St. Louis, Memphis, and Nashville Zoo, among the best in the country, as well as potentially one or two smaller places along the way, two national parks, and one or two renowned wildlife refuges.
One or two places could be cut out if you wanted to reduce driving or spend more time at one spot or the other. This itinerary is ambitious but easily doable without rushing in my opinion.
Are there specific zoo animals you and your son would like to see? Also, any natural areas or wildlife you are both interested in?