There are several race tracks in the Central Ohio area. Just depends on what you're interested in. 99% of the tracks are indoor. Here are some off the top of my head. I'm sure there are a couple more.
Offroad:
EN Hobbies(carpet) - Columbus
Ohio RC Factory(dirt) - Jeffersonville
Onroad:
EN Hobbies(carpet) - Columbus
Access Hobbies(carpet) - Springfield
Queen City RC(carpet) - Cincinnati
The Gate N.O.R.C.A.R(carpet) - Brunswick
Mansfield RC(carpet) - Mansfield
One Eighth Racers of Toledo(carpet) - Toledo *They also have an outdoor asphalt track in the summer*
Most of these tracks have Facebook and/or websites