I just stumbled across this thread, so sorry for the late reply.
I can't give you any advice about different hospitals as I lived in Nashville right after college graduation and didn't have a nursing degree, but I can tell you some about the Vandy/Hillsboro area. I lived right off of the exit for 21st Ave (the street Vanderbilt is on), in an older neighborhood. I was maybe 1 mile from Vanderbilt and close to the shopping district around the hospital (21st Ave. and West End Ave.). There were alot of coffeeshops, little local stores, bars (it's got three universities in that area, so there's alot of college kids.) That area is also close to downtown--it's astraight shot down 21st Ave/Broadway, and there was Centennial park (with the Parthenon you could visit), and once on West End you could go out to Baptist Hospital and other places pretty easily.
I loved the area (and Nashville in general because there was so much to do), though I rented (a walkup in a nicer historic home), so I can't say anything about the home market--though I know there were some crazy expensive houses for sale in the area (I saw one going for $880,000). Becky8, you can find some good deals for house/apts for rent in the Sylvan Park/Belmont/12th south area, but it's not nearly as nice as the Hillsboro side of Belmont University (basically Belmont's campus is sort of a divider for the nicer area close to Vandy and the 12th south area which isa bit....rough at times.) I'd say look at apartmentguide.com, Craigslist, and google for local rental places in Nashville. And if you want an estimate of rental ranges, a friend of mine paid $550 a month for a walkup one bedroom near Vandy and I paid $1300 for a 2bd walkup in the same neighborhood with all utilities included. IDK what your price range is, but there's an idea.
Oh, and if you're going to be driving from I-440 to Vanderbilt, you'll probably get off at the 21st Ave exit and go down that street through Hillsboro Village and the traffic in that area suuuucks. I could easily walk faster than alot of cars down near the medical campus because traffic was so slow.
Sorry I don't know much else about other parts of Nashville- I pretty stayed in that area. But if anyone has anymore questions about that area feel free to ask.