As an NYC expert (I've been there once, for 3 nights

- that's not quite true, I lived there for 2.5 years but was too young to remember it), I can only comment on my limited experience. I stayed at Hotel Lucerne on the Upper West Side (I now know what that term means

) and found it perfectly satisfactory, (guessing it used to be an old apartment block) and competitively priced fow what it was (though there are obviously heaps of options). It is only a couple of blocks from the American Museum of Natural History (I planned to go but did not get there), close to Central Park, and about a 30 minute walk down Broadway to Times Square. Also located close to the subway (I guess just about everywhere is).
Empire State building has great views but would recommend pre-purchase, anything to avoid the queues (and research the time of day you go, it was like a sardine tin when I was up there). The Rockefeller Centre also has an observation deck.
I didn't do too much else 'touristy' (I like to walk a city, and did plenty of that, around the fashion district, past Madison Square Garden, The Letterman show place, and the like) - went down to Battery Park and saw the Statue of Liberty from quite a distance and was roundabout Ground Zero - obviously a lot of construction work going on. Wandered through Central Park. I was there in Summer so things may be a little different around NYE! Loved it though. And depending on how much baggage you have the Airtrain (from Jamaica to JFK for $5, combined with a 2 or 3 dollar subway ticket is good value for getting to/from JFK). I posted elsewhere on making sure, if using supershuttle to/from the airport (which in itself is fine) that you have a printed confirmation 'ticket' - my driver tried to sting me again because I only had the number and not a printout (no printer available when I booked)
I'm sure there are people who know NYC much better than me who will chime in. Enjoy your trip.
PS. Though I only stayed 3 nights, there were either 12 or 15 'transactions' on my bill, all bar 3 being various taxes - make sure you know the full rate you're paying (I did, but it was still bizarre seeing all the extra txns on the bill).