We live in the same area as you're looking at. The schools you'll be zoned for will depend loads on where you live and its worth looking at elementary through to high school as you will find variations - i.e. elementaries can be great, high school may be not (I appreciate its a long way off for your 2 but worth thinking about!) all those areas are in different school districts to each other so its difficult to do a 'proper' compare
We've been here for 18 months, and were looking for 9 months before that. Overall the area / belt up the coast to connecticut is relatively well off, due to the proximity and ease of commute to NYC, and this in turns drives house prices (which drives property which drives how much is paid into local school and amenities taxes...) and the local areas
Out of the three, Rye is probably the most 'well to do' Has a nice little town center with small boutiques, a huge YMCA with lots of programs for the kids, lots of cafes, restaurants etc. However its VERY expensive house price wise. I don't know a ton about the school district, however I do know it has two (I think) Rye Neck and Rye and the two are separate e
Harrison I didn't love lots of houses and residential but missing a town centre (if that makes sense) for what we were looking for. Prices a lot more reasonable as a result, but also the school district at middle and high school was variable according to what we heard
Mamaroneck (our nearest town so I know more about it) relatively small but busy town centre with small independent shops and cafes and restaurants but not as 'naice' as Rye. Very Italian in parts which means awesome bakeries! They have a great town recreation department, a really nice cheap and accessible beach with huge playground by there, playing fields so lots of sports for the kids. School district is great - its a foundation with 4 elementary schools, 1 middle and 1 high school to serve the district, and its sought after. There's a massive swimming pool and ice rink at the local middle school that students can use and locals after hours. If I was buying again I would buy in Mamaroneck as it ticks all the boxes for us (we get getting gazumped as the market was insane so we didn't move thee)
Really all of the areas are nice, and all offer what you're looking for. When you say you dont want to drive, do you mean to activities or just generally or that you just want one car? I ask as every family I know here has at least 1 car - you have to drive to get to the supermarkets (no tesco express or similar here) school (if they don't provide buses - worth checking) Drs, after school activities etc and round here as its more spread out its a common theme that 'everywhere is a 15 min drive away' You will pay a lot more to rent or buy a house within walking distance to the station etc
Happy to answer any questions if you need more info