Check out Pearl River. It's only 32 miles outside the city, but it's like another planet - the epitome of the quiet little suburb with SO much to offer and super schools.
Lots of parades (the second largest St. Patrick's Day parade in NYS), festivals and carnivals throughout the year. Just a real lovely community. And there is little to no crime. The town motto is "The Town of Friendly People" -- and they live up to it!
I was raised there, and have many, many fond memories. From what I've been told, it hasn't changed that much over the years -- it just got more expensive. There was a time when it was almost all Irish Catholic, but that's not the case anymore - it's extremely diverse.
Oh, and there's the cutest little train station with service to the city, so that's a plus.
I think you would really like it - especially with kids. My opinion -- It's the best place in the world to raise kids. Safe, good education -- I wish I could afford to live there, I'd go back in a heartbeat!
Just found this link for you -- it's from their local paper - a photo tour of a Day In The Life of Pearl River. Enjoy!
Pearl River, New York