Hello. I currently work in the Biological Sciences department. Nashville is a pretty decent place to live. Some areas, can be really expensive to live in, such as Green Hills and Hillsboro VIllage. If you want or need to live really close (as in walking/biking distance) to campus, you'll pay a premium. Some of the cheaper areas to live in are West Nashville and Bellevue. I found a nice apartment in a really fancy neighborhood in West Meade for $800. It was only about a 10 minute drive to work. I now live in Woodbine and it take me 7 minutes to get there. Look for apartments and houses on Highway 70S (that's also the West side). You can find some cheap places in East Nashville and there are a few trendy bars and what-not, but be sure to visit the houses as some areas on the East side are still pretty rough. The Rosebank area of East Nashville is pretty nice, but it can take a little longer to get there as well. I'm not terribly familiar with North Nashville, so no help there. You'd probably do best to avoid places that are further out of the city, like Gallatin, Antioch, Smryna, and Murfreesboro. If you go to the Vanderbilt website and look under the tab for students, there's a tab called "MyVU" which has some classified ads that employees post offering places to rent. You'll find some pretty cheap places there, too. I don't recommend trying to make it without a car here, as our public transit system is pretty dodgy and the city is so spread out that a ten minute drive turns into a complicated, hour long journey. If you absolutely have to, it can be done.
Hillsboro Village is where most of the grad students can be found. This area is pretty nice with a lot of shops, restaurants, and bars. It's really close to the Belmont area, too, so there's quite a bit of overlap. Like I said earlier, there's a trendy area on the East Side called Five Points where you can find 3 Crow Bar, Red Door East, Alley Cat, and a record shop. Around the corner is the Lipstick Lounge and Mad Donna's. There's a few art galleries over there (Gallery East) and pub/art crawls aren't uncommon. This whole area is pretty cool and newish, as this is the area that was destroyed and rehabilitated after the tornado a few years ago. There's even a permanent hot dog stand called I Dream of Weenie.
I've lived in Nashville for a really long time, so if I'm glad to offer any help that I can. Just send me an email if you like! Wherever you end up in Nashville, you'll be fine. We're a very friendly people.