Other than friends, I've actually never seen anyone open carry in the area. No one looks down on it though from what I've seen.
Green River is the only place in the Bowling Green area I know of and it's a Gun Club, which means membership or friend with membership to get in. They sponsor weekend competitions, but I'm not sure how that works.
Sherwood Guns is a great place to get firearms but he specializes in handguns and doesn't keep store hours. Most of his contact is through phone until pickup with the standard instant background check unless you have a CCWL.
Just a block from him is Guns'n'More on Nashville Rd, but the owner charges about $100 over market for all firearms. And lately he's overcharging for ammo too. Decent selection of accessories though if you want to check it out.
There are a couple of pawn shops that deal with firearms, but I don't know much about them. There is Gander Mountain as well, but again they overcharge on most things and require trigger locks placed on your firearms (provided) while in the store if you don't have a CCWL.
I'm not aware of any local gun shows. Most of the shop owners around here head to Louisville and Lexington to sell at shows.
Also, if you ever want to go shooting with others, PM me for my phone number and I can set something up.