plenty of good BBQ in Memphis, all the recommendations above are good! If you really wanted to be touristy and see beale street (youll do it once and basically be over it) Blues City is good and is literally right on beale street
definitely stop by the bass pro, one of these days im just going to get myself a room in the hotel inside that pyramid it's awesome, definitely stop in
For those mentioning the Chuck Wagon in HSV, Love that place. at my previous airline we stayed right near it and I would always go, if it was open, they always had weird hours.
Still my favorite BBQ I've ever had is Southern Soul in St. Simons Island, GA (KSSI) (basically Brunswick, GA) its like a 2 block walk from the airport.
Also as far as airports go I spend a lot of time in Memphis, but I've only flown my airplane in once, and I flew right into KMEM and parked at Wilson Air, they treated me very very well for being a single prop guy, price was reasonable at $15 a night. I had a good experience and If i were to fly my plane in again I'd probably go right back to them