Nashville local here. At brunch right now and will respond more completely later.
Short version:
There is a nice 25ish mile (there and back) ride along a bike path and greenway that takes you to a dammed lake that you can hop on right outside your door.
There are a bunch of other rides around town you could do as well, but they require a bit more planning.
If you have access to a car, that opens up a lot, including the trace.
As for renting a bike, I’d check with Moab. It’s a quick Uber from where you’re staying. I think they rent.
I am a very slow rider, but if you can handle that, happy to connect.
Will