Hi there, Barcelona!
I actually have the opposite problem!
I will be moving to Barcelona with my soon 6 year old son and am right now in the middle of searching for good schools and neighbouhoods to live!
I am sure you could help me out a bit as you are actually from there
I lived in NY for two years and can recommened "Les enfants", it is a Montessori school in Astoria, Queens. It is very good and has kindergarten and all grades up to 6th grade. But I am not sure what kind of housing obtions there are if you would like to live nearby the school.
A good area to live with kids though is Riverdale, just above Manhattan. It is green and quiet with friendly people
The public schools there are also good.
Would be happy to talk more with you, maybe through email?