Lori and I were in NYC on vacations these last two years, and what you do in the city depends a lot on how much money you have. If money is not a problem, take cabs if you want to. They're really not that much more expensive, and if you're in a group, it costs the same for four people as it does for one. Subways are faster, but might be more of a problem with a carriage, as there are usually stairs.
I was never in any part of the city where I've felt unsafe in these last few years.
Central Park has playgrounds, but remember - it's bigger than most cities! Get a map. Lori and I walked and ran there a lot when we visited. Loved it. Also, don't think you'll want to be hitting the outdoor playgrounds with a baby in the winter.
NYC has everything. What are you interested in?
S4