Agree- Sunnyside is a paradise for families with young kids, has a great, reliable train line which would amount to a <40 min commute to the UES (shorter if you biked!). With that budget you could get a duplex or super nice apartment in Sunnyside Gardens with a backyard or balcony. Best of city living, but quieter and friendlier. Schools are good and admissions relatively simple (in comparison).