To get an overview of the city take the double decker bus tour. There are 2 basic routes up town and downtown. You can get on and off all day long. Most of the tour guides are pretty knowledgeable about the city. You can get a 2 day ticket that includes the Circle Line (boat trip around Manhattan).
Shows, museums, tours tons of things to do. How about a tour of Radio City Music Hall? See the stage, discuss the decor, meet a Rockette.
Tour NBC studios, Riverside Church, the UN, etc. Shows-on or off Broadway, concerts, music from opera and classical to pop and rock and everything inbetween.
What interests you? I'll bet that there is a museum dedicated to it in NYC! Police Museum, Fire Museum, transit Museum, South Street Seaport Museum, Metropolitan Museum of Art, Museum of Modern Art, American Museum of Natural History, Museum of the City of NY, etc.
www.nycvisit.com -website for NY & Company the convention & visitors bureau. They have a great site with links and a great office near Times Square.
www.broadway.com Broadway shows.
www.nystarlight.com Broadway and other shows
www.newyorksightseeing.com Grayline bus tours including the double decker buses.
www.timessquarebid.org The Times Square Business Improvement District aka Times Square Alliance They have a good information center in Times Square.
As for restaurants and eating one of my friends from California calls the menu prices in NY "Theme Park Prices". (You know the prices we pay at WDW for food.) But you can eat less expensively and well if you stay away from hotels and big name places.
What ever interests you have, you can find it in NYC.
Feel free to email or PM me for more info!