There are a number of great areas to raise a family around Chattanooga. I recommend the following zip codes: 37405, 37409, 37411, 37412, 37415, 37416, and a few more.
Here are some links to demographic data and listings for each of the areas listed above:
You can find more info on the other areas from that same website. You can find real estate listings by area under the Search Properties tab and demographic real estate info under the Community tab. Just click on an area to find out more.