Greenburgh Nature Center is located in Scarsdale, NY. http://greenburghnaturecenter.org/
It has a little museum, a trail, a pond, and some outdoor animal exhibits (goat, lamb, turkey, owl, balk eagle, hawk, rabbits, etc). They are also building a playground right now.
There are also many events that they organize for kids and families. For example, this summer they have open-air music series on Thursday evenings. And there are other events where you can have close encounter with animals.