Depends where in NY. A 2 bedroom outside of manhattan would cost at least $1500 a month and most likely tiny and roach infested. The farther you get from manhattan, the cleaner and cheaper they get (not necessarily cheaper, but more square footage for the money). $4 a day round trip to take the train, about $15 a day roundtrip if you take the railroad (depending on distance), $5 or 6 for breakfast, about $9 for lunch
childrens school fees i have no idea about. I went to public school all my life but I know of several private schools in manhattan that charge $30k a year. even for KINDERGARTEN! Public schools are mostly by zone, so if you move to a decent neighborhood (farther from manhattan), the less worried you should be. Food and other expenses vary from household to household. Outside of manhattan, i dont think groceries are too different from other parts of the country. I pay about $100/mo on avg for electricity, $165/mo for cable (2boxes with dvr, all premium channels, and cable modem), cell phone should cost the same nationwide i think, and car insurance is gonna cost more for sure.
I live in sunnyside queens (google map 11104) and pay $1200 a month for a tiny one bedroom with alot of roaches. lived here 2 years and heard rent went up a considerable amount since i moved in.