Big Onion Walking Tours is a really great way to see the city, if it's not too cold. They have descriptions of their tours and information on signing up at their website.
http://www.bigonion.com/
Don't be afraid to travel into the boroughs either, try Arthur Ave. in the Bronx for great Italian shopping
http://arthuravenuebronx.com/, Main Street in Flushing, Queens (take the 7 train to the end of the line) for a tourist free Little Asia, and Jackson Heights, Queens (take the R or 7 train to Roosevelt Avenue stop) to shop and eat in a great Indian community, or Greenpoint, Brooklyn (Metropolitan stop on the G train) for a Polish neighborhood.
If you are taking the subway, make sure to get a one-day fun pass for $4 at the machines at any stop (you can't get them from the manned booths) if you are going to ride more than three times. You can also get a free subway map at any manned booth.
Feel free to contact me off-line with any more questions!! As you can see, I have a bit too much to share!
Sheryl Woodruff
Curatorial Assistant
The Museum of Television & Radio