In order to carry a concealed pistol you need to take a course showing you have a basic understanding and level of skill with the weapon, but that's it. In some states open carry is allowed, where you can literally walk down the street with a loaded AK-47 sling over your shoulder and a pistol in a holster on your hip. Do a YouTube search for "open carry ak47". Something like that is pretty pointless, but it would be fun to see people's reactions.
But in order to own a weapon for home defense or hunting or anything where you are not concealing it, there are no permits or anything. It's our second amendment right to do so in this country.