Central Park (south side stinks of horse poo), time square (the famous NYPD department, Hershey store, M&M's store, bubba gump shrimp co), go see the financial district and realise how big 9/11 was, even just to see the area and the monument. Statue of Liberty, Macy's, stroll 5th, 6th, Park and Madison Ave. Bloomingdales, Rockafeller, Madison Sq Gardens, Empire State building. Grand Central station as well is just massive, get into the main terminal and check out the ceiling, it mesmerized me..
Walk 42nd street too, that's interesting. As is most of broadway and the surroundings of Time square. Try having some food at the Bubba Gump shrimp Co at Time Square too.
Spend a day around central park, if the weather's nice it's a lovely place to burn a whole day on. If the weather's good, just walk everywhere unless it's more than 12-15 streets away, as it's always nice to gawp at all the architecture.
Some good gigs up at the meat packing district if you're after music let me know. Also some good whiskey bars around too.
Stayed at the Waldorf Astoria and the Palace, both reasonable hotels and got some real history in them.
Take a map for your first time, but if you're walking around I bet after 2-3 days you'd throw it away.
Oh and I don't know if they've still got the rule that you can't take bags on the statue of liberty, so if you're going down there just take a bottle of water and catch the metro down, leave everything bar your camera in the hotel if so.
Last time I went I had to go because of work, NYC in the summer fully expenses paid was top