In Memphis, as suggested, visit Sun Studio, plus Stax Studio, and the Rock & Soul Museum for some great music history (and of course Graceland if you're into Elvis). The Rock & Soul Museum is in the Gibson Guitar Factory.
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www.sunstudio.com]
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www.soulsvilleusa.com]
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www.memphisrocknsoul.org]
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www.gibson.com]
I go to Memphis 2-3 times a year, great place; going again in about 6 weeks. As far as eating...BBQ! My favorite is Jim Neely's Interstate (his nephew's run a chain simply called Neely's too), but Corky's is fantastic, and Rendezvous is good for something different like the famous Memphis dry rub ribs (though technically not BBQ but grilled).
Great live music on Beale most nights, the Rum Boogie (and the bar right next to it...forget the name...but there's a backdoor between the two you can use) and BB Kings usually have very good live bands. Isaac Hayes also has a very nice bar/restaurant downtown at Peabody Place. Not sure about open mic nights at any of those places, but worth checking into.
In Nashville, must stop by Tootsies Orchid Lounge, probably the most famous honky tonk in the world. And for guitars, the world famous Gruhn Guitars is right down the street from Tootsies. And between the two is Robert's Western World, another pretty good honky tonk.
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www.tootsies.net]
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www.gruhn.com]
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www.robertswesternworld.com]
I do believe Tootsies might have an open mic thing on Saturday afternoons. But again, many places on "lower Broad" (near Broadway and 2nd Avenue) worth checking into regarding open mic nights. There are a few more places on 2nd Avenue and Printer's Alley (between 'lower broad' and '2nd ave') that might be worth checking out too.