For fresh fruits, vegetables, preserves, and honey, go to the Nashville Farmer’s Market held daily in the large covered building between 8th Avenue and the Bicentennial Mall [1]. Many of the goods for sale here are home-grown at farms near Nashville [2]; to be sure, ask whether you’re getting Tennessee products.
There is an abundance of traditional grocery stores around Nashville. Common chains are Kroger, Publix, and H. G. Hill. Drive out any of the main corridors into the city and you will quickly find a grocery store. There is a Trader Joe’s, which specializes in organic and specialty items, in Green Hills, just south of the Green Hills Mall.
Links:
[1] http://www.moon.com/destinations/tennessee/nashville/sights/jefferson-street/bicentennial-mall
[2] http://www.moon.com/destinations/tennessee/nashville