As with almost every question concerning gun laws, it depends on what state you live in.
What you are talking about are private party sales, where a guy sets up a table and maybe sells from his personal collection, and are not subject to any background check. Just a exchange of cash and a handshake and you are on your way.
Many states have outlawed this, but many have not. My state of California has outlawed this. Yes, you can buy from a private party, but the transfer must be handled by a license dealer, in full compliance with state and federal gun laws.
As for transport and storage, again, without knowing what state, I can't really help you.
UPDATE: Texas has the gun show loophole. Look for a guy with a table set up and look for a sign saying something like "Private Party" or something like that. This is a guy just selling out of his personal collection. An exchange of cash and a handshake are all that's needed to buy a gun.