You can also drive to the National Matches at Camp Perry during the summer. Lots of stuff to participate in there. West of Sandusky. Probably only an hour and a half drive for you.
Here's the CMP calendar page.
Here is the NRA's. Note the Army's Small Arms Firing Schools are on these schedules are one of the best bargains for shooters wanting a taste of competition. Also check the
CMP and
NRA for more local matches or registered leagues.