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The areas you mentioned are definitely a good start. My main advice would be to look in areas near where you intend to eventually buy. That way, you can get familiarity with the area, and you won't need to change your daughter's daycare abruptly.
Some other areas that could fit your bill:
Pleasantville -- very walkable village with excellent schools. Several day care options in the vicinity.
Mount Kisco -- a large walkable village. Bit more diverse. Schools are good, but less prestigious than some surrounding areas. But it would put you in a good location to explore the surrounding more prestigious areas, such as Chappaqua, Armonk, Briarcliff, Katonah.
Croton -- A walkable village, very good schools. Bit further north, but express trains make for a 45 minute commute.
Mamaroneck/Larchmont -- Both have nice walkable downtowns, with Larchmont being a bit more upscale. Schools are very good, but some people are put off by the size and diversity of the high school.
I would suspect you can find decent 2+ bedroom apartments in your price range, in any of those areas.
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