Nashville doesn't have a high and low season for apartment rentals as extreme as some other cities. In general, there will be more apartments available during the summer months. However, there might also be more people competing for those apartments, which makes aiming your search at the earlier portion of the season wise.
Realistically, if you're not in school or employed you will have difficulty renting an apartment. You should also be prepared to share the specifics of your education or employment situation with a landlord. They will likely want to know where you're going to school or working before renting to you.
With three major interstates that run through the city, most people consider car the fastest way to get around Nashville. However, the Nashville MTA bus service offers excellent routes downtown, and works very well for people commuting within the city center. For those with a longer ride, the Music City Star is a commuter train that runs on weekdays.
Entertainment in Nashville is centered on music. The Country Music Hall of Fame is a great way to get introduced the history of the music and the city. Attending the Grand Ole Opry is a way to be a part of country music history firsthand. The Mall at Green Hills provides a high end shopping experience where you can occasionally sport a country music superstar. Yazoo is the largest local craft brewer, and is well worth a visit. Most of the bars in the city feature frequent live music.