Just back from NY. Flew to Newark, taxi was I think $90 including tip into Manhattan. There's a train that goes to Grand Central too every 10 minutes which is cheaper (I think I read $12.50 each).
I would go against the grain and say don't stay very near Times Square or even in midtown - it's like going to London and staying in Leicester Square - sooooo many nicer places! Times Square is the type of place you go once to see, find it's rammed with people and there's not actually that much to do, and leave an hour later.
I would far prefer Greenwich village, Chelsea or Soho, personally. Places with a far more local feel with great variety of restaurants for the evening, as well as shops and other entertainment.
That said, what I find with New York (and I've been several times) is it is really surprising now small Manhattan is. So even if you stay in Times Square, hop on a subway or in a cab and you're down the other end in 10-15 minutes. I don't think it matters where you stay nearly as much as it does in a larger, more spread out city.
On that subject, we thought the subway was great, cheap, really easy to use and much more compact than London. It didn't seem to take any time at all to get anywhere. Most of the time you just go down a short set of steps, swipe in (get a free metrocard from the machine which you can top up any time easily) and you're right there at the track. Cabs were not expensive either - we got 30 blocks in midtown for $7 (plus tip...) And you can walk 10 blocks in about 10 minutes.
As others have said, I would definitely not bother with hotel breakfasts. I LOVE an American breakfast menu from a deli/diner (French toast, pancakes...) - probably better food and less expensive than a hotel breakfast.
Things to do...we spent half a day at the 9/11 memorial and museum, which is downtown. I would recommend. Same end of town, go over to Brooklyn on the subway, have a look around, walk the bridge (maybe a mile) back for amazing views. Wall Street is just beside the memorial and museum, if you want to have a look (it's just offices though!) If you/your kids like designer stuff, Century 21 is right there too, it's the biggest designer outlet in Manhattan.
Again, in the same very small area, you can either get the ferry to Liberty Island, OR you can get the (free!) ferry to Staten Island which goes past the statue. It might be worth a trip to Liberty Island for your first visit, but we did Staten Island, and it was great - you get a nice seat and can get photos etc, and of course it's one of the few things that's free! It takes about an hour there and back in total, brilliant views of Manhattan on the way back.
Museum of Natural History on upper west side, lots of other galleries/museums on the upper east side if that's your thing.
I love an open top bus tour to see everything and get bearings...you will get accosted in Times Square with many offers!
Top of the Rock fantastic views, you get to actually see the Empire State Building rather than being up it. Go at sunset (book tickets beforehand).
Can you tell I want to go back?! Enjoy yourself!