I grew up in northeast ohio, but now live in Columbus. NEO has way better scenery, but basically nowhere to /out/ overnight.
I'll correct your mild retardation to let you know that in central ohio, you're going to head southeast to go anywhere and you have your choice between State Parks and National Forest. Wayne is a National Forest, but it is also surrounded by a few State Parks like Zaleski. If you want a real sort of /out/ experience, I'd recommend Wildcat Hollow in the Wayne National Forest. It's a near 15 mile loop and you have to cache water on the way, etc.
The State Park system is 90% focused on RV'ers and you're going to have a mediocre time at Burr Oak or Tar Hallow which are the other closest places to go. Last time I did Tar Hallow I had a lesbian couple and their autistic kid setup their camp 5 feet from my fucking hammock even though they had a 150 sq ft. spot to themselves.
Wayne National Forest allows you to go fully dispersed... meaning, head out on a trail, find a place to setup, and just camp there. State Parks require reservations. National forests are also really more chill about where you set a fire -- in state parks you do it in your fire ring or a ranger will fuck your shit up.
I do have to say however, you should probably do a state park this time of year if you've got no clue. It gets pretty cold at night already and car camping is going to be your best option else you might just fucking freeze. I don't care how much you think your bag is rated to -20, if you're not used to being out in the area, car camp it.
It's not terrible to car camp it at Tar Hallow or Burr Oak though, they have some lovely trails for day hikes. Just bring your own firewood because the lumberyard shit they'll sell you burns out really quickly.
