I live in Nashville and I love it here. I'm an old kook and not very good, but I love to skate when I can. Like mentioned above, there's 6th ave (indoor park) which is cool but is super slick. I rarely go because it's so freakin' slick, (everyone calls it Slick Ave here). Two Rivers is an outdoor park with two concrete bowls and a flow area with a few humps, rails, steps and so on. You can usually skate there without a helmet or pads, but I have heard stories of cops hassling people without a helmet (but I've never seen it.)
Hunt supply is a good place to get more info. It's a little hidden in an alleyway though and kind of hard to find sometimes. Another good shop to check out would be Franklin Skateshop 1227 Lakeview Dr #4, Franklin, TN 37067. The owner, Donny Myhre, was a pro for Zorlac back in the day. There's a mini half in the back room too. He's a great guy and can get you hooked up with local skaters.
Another shop to check out is Purpose Skate Shop 2546 Lebanon Pike, Nashville, TN 37214. The owner Dan is a skater through and through. The shop is small, but he has good stuff. It's close to Two Rivers skatepark. Dan can get you in contact with some local guys as well.
If you have a YMCA membership, I highly recommend checking out the Brentwood YMCA on Concord road. Their skate park is the 3rd largest YMCA skatepark in the country (according to the coordinator). They have a 12' vert ramp and huge wooden bowl and a street course with ledges, rails, euros and quarters and it flows well. Helmet is required. If you aren't a member, I think it's $10 to skate all day. It's my favorite spot.
There are a ton of other spots. Downtown is always a good spot for street skating but watch for cops. I hear they can be tough at some of the state buildings. There are lots of other spots and parks. If you stop in at any of the shops above they can give you more details. Welcome to Nashville!