Spring is the right time to walk the neighborhoods on your shortlist, because you can see them in use. Nashville has plenty of places to do that. Centennial Park, the Shelby Bottoms greenway on the east side, Radnor Lake south of town, and the Warner Parks to the west all fill up once the weather turns.
What you are really doing is gathering information a listing photo cannot give you. Walk the block in the evening and again on a weekday morning. Note whether people are outside, whether the sidewalks connect to anything, how far the nearest grocery is, and what the street sounds like at rush hour rather than at midday.
Look for the slower signals too. Remodel permits, small businesses opening rather than closing, and parks that are visibly maintained all suggest a neighborhood people are investing in. County permit records and school attendance boundaries are public, and are worth checking against whatever you have been told.
Once you have two or three candidate areas, compare them on the numbers as well. Property tax rates and insurance costs vary between jurisdictions, and the difference can outweigh a modest gap in asking price. The calculator at /mortgage-calculator/affordability/ will show you what that looks like month to month.