I haven't checked in here for over a year, but was looking for some other information and saw the "newish" comments on this thread. I'll give you a quick update so that it might help the two of you who are moving in January if you still need the info.
Where to live: This is not too "country" at all. It is really just the suburbs of NYC. Comparable to Surrey IMO.
If you want a little bit of walkability, I think Old Greenwich/Riverside are the best areas to be in. There are a lot of expats here. Mostly British/Irish/Australian and South American (Argentinian/Brazilian). There is a small parade of shops in OG and the beach is very close (bike-able if not walk-able). There are several local pre-schools that are good (registration has just passed, but places may still be possible). Old Greenwich school and Riverside school are both good. More walkers at OG and more car drops at Riverside, but both communities are sociable. The school debate goes round and round (as in London), and there probably are better schools a bit further North (Darien/Westport etc.). But the commute is further and really, I'm not sure it makes that significant of a difference.
One word of caution-rent is very very expensive here if you want somewhere nice. And you will need a car. Possibly two cars depending on proximity to the station.
Highlystrung-the only issue that you might find with living in this area is that it's a hike to Whitby, especially in the winter. I've heard that people really like the school, but they all complain about the driving (someone as recently as Saturday at the playground...) There are some nice parts of Greenwich closer to the hospital and private schools in town that might be closer for you. Anything off Lake Avenue or North Street would be an easier drive to Whitby. Also, your kids will have friend all over the place and you will have to drive them everywhere...just a consideration.
It can be Stepford Wife territory here, the stereotype exists for a reason unfortunately. There are also lots of nice families. Just be prepared to have to work a bit to make friends. There is a newcomers club that we didn't join but other friends have joined and enjoyed it.
There are a million activities for children. Anything your child wants to do will be available. It is fantastic from that perspective. The facilities and programs for both boys and girls are top notch. These are lucky and privileged children.
A child who is 6 years old (summer birthday born in 2010) will be in first grade in the 2016-2017 school year. They do allow kids born from Sept-Dec to be held back, so there may be children who have turned 7 already in your child's class. They won't hold a summer baby back here as the intake generally runs from January-December. And holding back Sept-Dec children is entirely at the parents discretion. It's a bit mad really as my daughters class has both 7 year olds and 5 year olds in it.
I would say that it takes a year to settle. This is just my personal opinion, but it seems to be what has happened for us. Best of luck!