The best route to Nashville is by car. The town is extensive and so are its attractions. If you're not driving down then renting a car would make sense. That said, Nashville does have a public transport system, the MTA, but it is not entirely tourist-friendly.
MTA has buses which run through most of the city's areas. Circulator busses in the downtown area, called the Music City Loop, also travel through the city and provide free Monday through Saturday transportation. There are plenty of taxis available, as are Uber and Lyft cars. You can also take a pedicab-suitable for up to three riders, and this is a great way to experience Nashville for visitors.
infographic by: www.way.com