New Canaan has a wonderful spot for small children – Mead Park. There is a fenced a toddler play area as well as a more advanced playground, swings, and picnic benches. Additionally, there are several fields, over a half dozen tennis courts, and a lovely duck pond. To top it all off, there is a snack bar that offers an impressive variety of food that is quite yummy, tables with umbrellas, and clean restrooms.
I believe the tennis courts require a town permit to be used but the park seems to otherwise be open to the public for everyone to enjoy. You are also walking distance from the center of town.