Some tips: I've found that it takes 45 mins to get from the airport to the city, so budget accordingly.
As far as gettnig around the city, I've only used the Subway once so I can't give too much advice on that but the taxis are very easy and convieient. You can walk from Rockefeller Centre to Trump Tower (I acutally met and got a pic with Trump in the lobby last year-so you never know!). Empire State bldg is about a 10-15 min walk from Times Square. Don't hesitate to flag a cab, but enjoy the walk around the city.
I would def suggest checking out the Trade Centre site. JUst take a cab down. You can also hit the Statue of Liberty when down there. The WTC site is def a must see.
I would also suggest the New YOrk Sightseeing tour. It's the red double decker bus, get on and get off as you please. About $25...it's great for a first time visitor to see a lot of things.
As far as eating, 2 places that stuck out for me was Ruby Foos in Times Square (if you like sushi) or Angelo and Maxi's on Park Ave (great Steak House). But since you will be tight on time, I would suggest something quick. There is a great Deli right beside a Fire Hall down by the WTC, and another great one on 46th(ish) right beside a TGI Friday's near Times Square. There are lots of places to eat, you'll find something you'll like. And there are over 100 Starbucks on the island, so enjoy.
have fun.