There is a large system of greenways and designated bike lanes in Nashville and the surrounding areas. Look online for individual maps of each area. A map of the whole system. One of the Warner parks (Edwin or Percy) has a nice climbing loop frequented by cyclists.

SW of town, around Liepers Fork, west of Franklin, there are a lot of low-traffic rural roads with rolling hills to ride.

And there's always the Natchez Trace, with its terminus also SW of town.

Caveat - I have not had a lot of time to ride there. I always have good intentions and have scoped out some of these, but other activities have always gotten in the way.