Welcome !
Definitely check out your state laws. IMO, if your state allows OC, it's highly probably it will be restricted on public transportation; however, carrying on your bike seems logical...however, since when were bureaucrats logical !:lol: Be sure to check to see if local municipalities can restrict carrying too.
Since I've gotten back into cycling, I'm sorely tempted to strap on over my cycling gear. I'd prefer a mid-frame all-weather holster, but have yet to find one. What I've found is that most road cyclists are strongly opposed to self-defense. That might be because the throw themselves in front of much more massive vehicles on a regular basis. Many mountain biking cyclists, however, are more amenable to carry.
You cannot carry a loaded handgun in a vehicle without an Ohio CHL or recognized equivalent.
On foot, you need no CHL (or equivalent) to open carry a loaded handgun. In fact, you don't even need any ID.