In what state court would you file a discrimination lawsuit in Ohio?

It really depends on who you are suing and how much you are asking for in damages. For instance, if you're suing the state of Ohio, then the Ohio Court of Claims would hear the case. If you're suing a different entity than the state, Ohio County Courts hear most civil claims for less than $15,000. If you're asking for more than $15,000 the Ohio Court of Common Pleas would likely hear the case.

Because your question is not very detailed, I'm linking you to the Ohio section of a great court reference site. You can find lots of general information about the structure of Ohio courts, as well as specialized information by county.