Check out
GreatSchools.net and
SchoolMatters - Home for elementary thru high school ratings and other info. It is difficult to compare schools in two different states, but I have heard that the top tier Westchester Schools are better than Greenwich schools and that most people that can afford to in Greenwich send their kids to private schools. Since you live in Greenwich you are in better position to determine if this is true.
Unfortunately, $1.2m does not go very far in Rye but the property taxes there are better than most in Westchester. See
Houlihan Lawrence Real Estate for an idea of what $1.2m will get you in Rye. If you are looking for diversity I will recommend adding Larchmont (Mamaroneck school district) and Mamaroneck (Mamaroneck and Rye Neck school district) to your list, you will probably get a little more for your money there too. A part of Rye shares a school district with a part of Mamroneck (Rye Neck), but you can't walk to most places apart from that part of Rye except for the private beaches located there, Rye Golf and Pool and the Middle/High schools.
All three communities have great activities for kids including preschoolers and all three have a beach that is easily accessible to residents. Rye has an Arts Center and YMCA, but you don't have to be a resident to enroll your kids in their programs.
You will need to live close to downtown Rye to be able to walk to most places, Larchmont is much smaller so you are more likely to find a home from which you can walk to everything there. Mamaroneck is also walkable depending on where you live.
With regards to preschools in the area (Larchmont, Rye, Mamaroneck)...most are located in a church or synagogue even if they are not religous. Examples are Mamaroneck Community Nursery School (located at United Methodist Church), Little Flower Nursery School in Mamaroneck (located at Holy Trinity Church), Westchester (Jewish) Day School in Mamaroneck, Larchmont Avenue Church,St John's in Larchmont, Larchmont Temple, Resurrection in Rye (Montessori) and Rye Presbyterian Nursery School.