I'd highly recommend the St. Matthews, Crescent Hill or Highlands/Bardstown Rd area. I just looked on craigslist and found several homes in your price range in those areas. Most homes around there, particularly in Crescent Hill and the Highlands are older (many built in 1920's-1930's)-many beautiful streets with mature trees and well-kept houses. Those are also among the most convenient areas in the city for shopping/entertainment and are pretty close to downtown, with St. Matthews having 2 good shopping malls (Mall St. Matthews and Oxmoor Center, although Oxmoor isn't technically in St. Matthews).
If you plan on putting your children in JCPS the areas I mentioned would be your best bet for the better schools, although unlike most districts your children may not automatically be enrolled in the neighborhood school (they could be bussed somewhere to the other end of town which stinks but that's the way the system works, although that mostly happens with elementary-age students); for that reason you may want to look into private schools.
If you do indeed move here and have any questions about the areas I mentioned or specific schools, please ask

Louisville's a great place and no, I'm not just saying that because I live here.