The link I provided goes into much greater detail but basically the law is that you may carry a sidearm as long as it is in plain view (barring any local ordinances) and you are not prohibited from legally owning a firearm.
To legally carry a concealed firearm you are required to take the proper safety course, the KSP does a background check, and barring any problems you will be issued a Concealed Carry permit. The most the course can cost is $75.00 and the background check is $60.00. The turn-around from the time you mail in the application until you receive the permit is normally around 90 days.
Kentucky is a 'shall issue' state which means that as long as you meet all the requirements you will be issued a permit. As was previously stated, most government buildings are 'gun-free' regarding the private citizen. Businesses may post on their storefront a message stating that they request that you not carry in their place of business. Ignoring the message isn't illegal but they can request that you leave.
Gunshow purchases: if it is an individual-to-individual transaction no paperwork is required. If you are buying from a licensed dealer, he WILL require you to fill out the paperwork and they WILL do an instant background check. If the dealer isn't equipped to do the check on the spot he will require you to wait to pick up your purchase on the next business day, after the background check is performed. Most (if not all) gunshows in the state require you to be a licensed dealer to set up a booth.
FYI: there is no longer a federal 5-day wait on handgun purchases.