"Better schools" pretty much correlate directly with "pricier neighborhoods" in NYC. The elementary school in my neighborhood is fine, but I plan to be actively involved in my 3yr olds education. I don't love the situation, but I'm not too worried about it.

At the high school levels there are the magnet schools, which your kids could potentially attend regardless of the neighborhood that you live in, but getting into those is somewhat merit-based.

It's a pretty massive lifestyle change, but there are crazy positives. Everything I need is within walking distance of my apartment. When my daughter goes to the playground, it is packed with other children for her to play with. There's a great library system, and there are plenty of child-friendly events to attend. From a professional standpoint, there are always meetups and events going on that are but a subway ride away.

I used to live in NJ and never thought I would like living in NYC, but now that I have a daughter I can't think of any other place that I'd want her to grow up in.