One of my favorites is Gilson Park in Wilmette: http://www.wilmettepark.org/gillson-park. It's a free beach next to a park with plenty of space for a picnic and free parking. The lot closest to the beach is residents-only, but you can park on streets throughout the park for free, and you'll be at most 100 yards from the beach.

The park is next to the Bahai Temple, which is a place worth visiting, too.