Nashville Farmer's Market
Mar 25 is going to be early in the season to see the market in optimal conditions. If you visit the web site for the market right before you come to town, you can see if anything special is going on. http://nashvillefarmersmarket.org/ The site also has good general info about the market. Special events are almost always on Saturdays, though, and many of the truly local merchants aren't at the market every day even in the height of summer.
One thing to know about the Nashville market is that it's a market in transition. For many years there were few actual local farmers, lots of resellers, and a huge tacky flea market. It's getting better, and there are more real farmers (in a special section) and less crap. There's a nice garden shop on site, and inside the market building are some fun places to eat, as well. I'd recommend timing your visit so you eat lunch there. Still, there's just not going to be a lot of local produce that time of year -- take a look at the seasonality calendar on the web site to see what I mean.