I am not a lawyer and you should consult one who is well qualified in this area of the law if you want to know what to do and what not to do.
That being said, here's my take on it.
You may not carry a loaded firearm in a vehicle without a CHL - and even then there are onerous rules. Without a CHL, the gun and the ammo must be carried in separate compartments not able to be accessed without leaving the vehicle. (ie gun in a case in the back seat and ammo in the trunk)
Ohio has no state law against open carry. There are, however, many local laws that prohibit it. H.B. 347 which is under consideration
right now would end this patchwork of more restrictive gun laws.
Even though there is no law against open carry, if you are carrying a firearm in a public place where one wouldn't normally hunt wild game, you can expect to be hassled by police when someone calls in a "man with a gun" complaint. Though technically you are not breaking any law, you could easily be charged with disturbing the peace or inducing panic or some such B.S. For most Ohioans, the hassle isn't worth it and open carry is all but non-existant.
I should also mention that you must meet certain requirements to possess a firearm under Ohio law. You can find this information at
www.packing.org or at
http://onlinedocs.andersonpublishing.com/revisedcode/ under Title 29, chapter 23.
I hope this answers your question.
-Dave