I think greenwich is better than Stamford but I know more about Stamford ones being worse (well, its all relative) I had a friend who bought a steal of a house there as schools weren't considered great.
The school ratings site thats usually quoted on the rental / for sale sites is usually pretty reliable (altho a lot of it is based on test scores so schools will usually score lower if they have a higher % of kids with English as a second language)
As you come down the line towards NYC, Rye is a good school district (confusingly its split into two tho - one is better than the other iirc) , Larchmont / Mamaroneck is excellent, Scarsdale and Eastchester are also sound, Harrison is ok. Worth noting that elementary schools can be great, high school not so much but I view it as a long game - i.e. you may have a few years before you need to think about it. All of these places are (imo anyway) expensive to live -we quite often see 3-4 bed houses in a good school district renting for ~$6000+ a month atm
The car thing is not so bad once you get here - everyone drives like they've never taken a driving lesson in their life so its a pretty low bar
a 4WD or AWD is key tho for two reasons 1) the roads here are AWFUL - patched with massive craters all over them, not to mention raised manholes / stupid metal covers on roadworks 2) winter - it can snow a lot. This last winter went on for ages, unlike the Uk where it all grinds to a halt with 1-2 inches, you're expected to just get on with it and drive places. A decent car is key!
Might be worth you contacting an estate agent who deals with international ppl a lot - this side of Westchester they are used to it (have a TON of French people, lots of ppl who work for the UN etc) so can help narrow down with you what you want and your criteria and start honing in on areas that might suit.