If you are looking to move to Chicago one of the most popular neighborhoods because of the schools is North Center. North Center has great restaurants, shops, parks, etc for the kids and is centrally located to Lake Shore Drive and 90/94, and close to the CTA Brown Line.
Please feel free to call me to discuss your needs and wants in your new home.
There are a few great neighborhoods in the city and plenty of suburban areas that will provide you with you have outlined here.
Would love to have a one-on-one conversation regarding your specific needs.