I'm not local, but I looooove Memphis. These are Memphis suggestions.
Unless you're vegetarian, you have to get a cheeseburger at the Lamplighter. Last time I saw Jack O. playing in LA he was talking about the Lamplighter cheeseburger.
I personally love Wild Bill's, a juke joint on the other side of the tracks but very welcoming. They sell beer by the quart. Don't sit off to the side, get in the mix.
Aside from all the great music spots (Stax, Graceland, Sun) I HIGHLY recommend the Civil Rights Museum at the Lorraine Motel. Definitely a different gear than all the fun of Memphis, but a very moving experience. I was overwhelmed.
I'll let other folks handle the bbq suggestions.