Welcome to Ohio! My old stomping ground
You've got a good number of places to ride near you in central OH (I'm assuming Columbus)
Alum Creek has some OK trails and is north of you
The pics show the trails in the various areas and link to more info.
To the south you've got some super options.
Hocking Hills - Has a great cave that you can ride into
East Fork - 70 ought miles of trails - This was were I rode mostly
Paint Creek - Good scenery but recent vandalism has ruined the settlers village with log cabins and barns.
Caesars Creek - Lots of good horse trails on the west side of the lake
Hueston Woods - Another good place to ride but not a lot of mileage