I guess it depends on what you call "moderate" housing, but I would recommend the Edgehill area. Close to Vandy, bikable to downtown (you didn't say where you worked). Close to several city parks and Edgehill Villa, 12South, Hillsboro Village, West End/Centennial Park area for fun stuff to do. Hardware store and grocery store in Hillsboro. But house prices have not skyrocketed like some of the other neighborhoods yet. If the Amp goes through, anywhere on the West End corridor should get better for commuting by public transportation.
I agree with Mom to 5 that being close to Vanderbilt is great for a car-free lifestyle. And Mustachianism in general. Even though housing prices are higher than in other areas, Centennial Park and Vanderbilt have a wide array of fun and free things to do.