I am a Realtor on Staten Island, your best bet would to post this question in the manhatten section of trulia voices so you can get answers from people who live in the area your interested in moving to. If you would like to link up with a Realtor in Manhatten you can contact me directly and I can assist you.
Good luck to you and your family!
I live in Manhattan and I am a broker but I have been living here a long time..( I did recently adopt a shelter dog who came from Staten Isalnd!) I think first you need to decide what part of Manhattan you want to live in, what your budget is and whether you want to rent or buy. Where do you work and what kind of lifestyle and /or hobbies do you favor? ie: Do you like to eat out or eat at home? Do you like to go to the theater or play sports? Do you have pets? You might look into Meetup groups (http://meetup.com) in the areas you want to live in that relate to your hobbies or interests as a way to meet other families in the neighborhood. The Upper West Side is a wonderful if somewhat expensive neighborhood situtated between 2 parklands with many restaurants and amenities..(there are some details and links on my blogs and website)
Please contact me if you would like me to represent you as your buyer broker at additional expense!
As for moving to Manhattan with a family, it all depends on what part of Manhattan and will you be renting. I'm sure you are aware that the prices in the city are much higher and you do not have as much land for the kids to roam.
You will also want to look into schools, you didn't mention whether your children would be going public or private. Here is a link to research public and private schools in New York : http://www.greatschools.net/modperl/go/NY
I do not have personal experience living in Manhattan but I know a family who lived on Sullivan Street and raised two children there. There kids grew up a bit more street savvy and seemed to be cultured due to the diversity and artistic happenings they were surrounded by.
Good Luck to you and your family.