1. Go to Central Park - it is one of the largest and most beautiful parks I've ever seen.
2. Just walk around, buy street food, and take the subway - New York is such an awesome place that you'll find lots of interesting things.
3. Go to Times Square (it's awesome!)
4. Go to New York's Chinatown - I've been there, and you can find really cool stuff and really unique and delicious food.
5. Visit the Brooklyn Botanical Garden - it's free Tuesdays.
6. Go to 5th Avenue - it has tons of cool architecture and stores. 5th Avenue includes the world's only NBA store.
7. Some architecturally amazing buildings are the Empire State Building, the Grand Central Terminal, the Rockefeller Center, the Solomon R. Guggenheim Museum, and the Seagram Building
8. Visit the Sony Wonder Technology Lab. It has free admission.
9. New York's amazing and often gigantic cathedrals are also fun to visit. Some good ones are the Cathedral Church of St. John the Divine, the Riverside Church (has the world's highest carillion tower at 400 ft), and the Gothic St. Patrick's Cathedral (also on 5th Avenue)
10. The New York Public Library is also amazing
11. Take a cruise on the Staten Island Ferry (it's free) - you'll be able to see the Statue of Liberty from it!
12. Walk across the Brooklyn Bridge's footpath. The view is amazing! Then you can walk along the Brooklyn Heights Promenade.
13. Visit the Solomon R. Guggenheim Museum, MOMA, and other museums.
Hope that helps!
I've been to NYC a bunch of times.