I would look in Monrovia, which is not a town but is part of Madison County. It would be NW of the Arsenal. Commute with traffic would be 15-20 minutes in most cases. There are homes in all price ranges in Monrovia and lots of new construction. Go to
www.valleymls.com and enter your price range and narrow the search by entering Monrovia Elementary School or Legacy Elementary School (further out) or Endeavor Elementary School (even further 20-25 minutes in morning traffic a bit more in afternoon). Sparkman in the high school.
You could also check in Madison, AL. It would meet your time requirements and school requirements but it will be a bit more expensive. Your money will generally go further in the County.
Monrovia and Madison are both residential "suburbs" of Huntsville but Madison is a small city and has a small downtown area and a nice sense of community identitiy. Quite frankly, I don't consider even Huntsville to have a "city feel" but that would be a personal preference.