Everything really depends on your budget. NYC is a great place to visit, if you have the cash to spend. It's expensive - everything
costs more - meals, hotels, a bottle of water, you name it. One of the best ways to save money is to use the subway and not spend money on cabs. Get an MTA map and memorize it before you leave.
As far as where to stay, Midtown is really the heart of Manhattan and will put you right in the middle of everything (other than attractions downtown, such as the Statue of Liberty, the 9/11 memorial, etc). Unfortunately, Midtown is also the most expensive area, so plan accordingly. Hotel rooms in Manhattan are small (DO NOT be fooled by photos on hotel websites). Claustrophobically, horrendously small, so be prepared for that. You might be better off doing a short term condo rental that would accommodate all of you rather than bothering with a hotel. You could also save a lot of money by cooking some of your meals rather than eating out.
Despite anything you may have heard, NYC is probably the safest big city in the US. There are cops everywhere, and really, unless you are wandering around Alphabet City at 3:00 in the morning (doubtful, unless you were interested in buying crack or meth) you can walk anywhere in NYC and not worry. That doesn't mean the standard rules of tourism shouldn't apply, they do and you should use the same amount of caution that you would use anywhere else.
Take advantage of as many free things as you can. Central Park, for example. A great place to let the kids run around and let off steam. One way to save money is to is visit the TKTS booth
in Times Square where you can buy same day theater tickets for half price. Just be flexible on the show and your seating arrangements. But half price to a Broadway show is a bargain.
As far as paid attractions go, everyone yammers about the Empire State Building, but my personal favorite is the top of Rockefeller Center. The view is much better, and you get a nice tour presentation to go along with it.