The main issue here is which school are your children going to be attending and are they all going to be going to the same school. Private school tuition runs between $35K to $50Kand more, annual, per child plus textbooks and plus after - school activities.....depending on which class they're in. Transportation has to be provided between point a and point b...are you going to have a nanny to pick them up?
You would have to fulfill an application and be accepted by the authorities at the school; it is quite a long process including interviews. I can recommend a school coach who helps with relocation customers if you like, who can walk and talk you through the process..
As far as three beds go, usually, they run anywhere between $4500 for a flex three (min) (meaning a wall is cutting off the living room in two-with landlord permission...-not all landlords allow walls) to upwards of between$6,500 to $15,000.00 in a doorman building. Tenement buildings which are walkups are cheaper, but the rooms are tiny.
I currently have a listing for a $10,500 three bedroom on the upper east side which I can send you if you give me your email address; everything brand new renovated. Keep in mind landlords are looking for income equal to 40-45x's the rent and substantiated documentation in the form of credit check, tax returns and accountant letters, as well as bank statements, reference letters etc... have to be presented for your application. Not all buildings accept guarantors - there is also Insurent which is a company that acts as guarantor for a fee. (not in all buildings).
If this sounds a bit difficult, there are other parts of the city that are less expesive and one area I would recommend is Forest Hills (Queens) that offers larger apartments and has good public schools.
Let me know if I can be of further assistance.
Licensed Real Estate Salesperson
Metropolitan Property Group
255 West 36th Street (Suite #406)
New York, NY 10018