Hey, L. I am currently an undergraduate at from the DC suburbs. It definitely has a small city feel compared to Boston, New York, or DC (out of curiosity, where are you from?) The traffic really isn't that bad here compared to other cities. However, you'll definitely want to live around campus. The good news is that rent is fairly inexpensive. Where you can live will vary depending on the amount of your stipend and how much you want to allocate to rent, of course. I would suggest the Belmont area, West End, or Green Hills. You can't really go wrong with those neighborhoods and they should give you some flexibility. Do a Craigslist search to get an idea for what they have to offer. I found a 2br for $650. With a roommate that's not bad at all. There are a few trendy neighborhoods that are "up and coming" like parts of East Nashville and the area around 12South (on 12th Avenue South near Belmont) if that's your thing.
It's good that you are bringing your car. There isn't a great mass transportation system in Nashville. They have bus service but it probably won't be helpful to you and Nashville definitely isn't a walking city. However, like vbrooksutk said you can easily bike from university areas to save some yourself some gas. There are tons of cool restaurants, bars, coffe shops, art galleries, trendy shops, and mainstream retailers. Within a 5 to 10 minute drive you can get Thai, Indian, Middle Eastern, Greek, Italian, Japanese (sushi and other stuff), etc. Most of those places are within walking distance, too. There's Starbucks but there are a ton of local coffeshops that are a great place to study (a couple of them are frequented more by graduate students) and have much better coffee. It definitely feels like a cosmopolitan place with a little southern charm (I'm a big meat and three fan.) Anyway, I could go on and on. It's not a bad place to be a grad student. What program are you thinking of joining? If you have any other questions, feel free to drop me a line on this board if you have any more questions about Nashville in general.