Well those are certainly very different areas! Lincoln Park is very urban. It's a long walk or short bus ride to the magnificent mile, but parking is very scarce unless you pay for a garage spot. It's a very exciting area to live, especially for young & wealthy. That means Chicago Public Schools or private. Chicago public schools are very hit or miss, plus there are magnet schools to complicate the picture. If you want to live there, I would ask the realtor in the area which school is the attendance area school, then check it out at this site:

research.cps.k12.il.us/cps/accountweb/

You would also want to talk to someone in the the same area and tour the school. One tactic might be to strike up conversations with parents in local playgrounds.

The two most prestigious private schools in the area are Francis Parker and the Latin School. But there are certainly other private schools as well.

Oak Park is an older suburb with a very diverse population, ethnically and economically. The population tends to be liberal, and the neighborhoods have front porches and sidewalks. I don't know much about the middle schools, but based on the type of people who live there, I assume that they are good if your kids are typical learners. I've heard that the district is not the very best if your kids have learning disabilities but I really don't know first hand.

Hinsdale is also an older suburb, but it's a bit pricier. I know nothing about their schools, but I assume that they're excellent just because the area is expensive.