There are bikeways all in and around Nashville.
Here's a link to maps and info:
https://www.nashville.gov/Public-Wor...y-Bikeway.aspx
and another site:
https://www.walkbikenashville.org/maps
My daughter lived SW of Nashville for a while out in Liepers Fork, and although I didn't get a chance to ride there, the area surrounding that looked very attractive to riding. - nice scenery and lack of auto traffic. Of course the Natchez Trace is out that direction, too.
I don't know that much about the riding on the north side, though.