My recommendations:
Touristy: Walk the
Brooklyn Bridge, and time it so that when you get to the other side you can have some of the best pizza in NYC - Grimaldi's Pizza. Grimaldi's is right on the Brooklyn side of the Brooklyn bridge and it's really great. It is very, very busy at peak times, so try to time yourself there for a lunch rather than a dinner, and don't even bother with dinner there on the weekends - too crowded. Website:
http://www.grimaldis.com/2/Index.htm
People Watching: I don't think it gets better than
Union Square for people watching, personally. There are other great areas of town to people watch, for sure, but I prefer union square. Try visiting union square and then having lunch/dinner in the area - tons of great restaurants. The mac n' cheese at chat n' chew was mentioned, and I can second that. But it really depends on the cuisine desired. You can literally find ANYTHING within 5 blocks of union square.
Nature-y: Many people would suggest a walk through
Central Park. I certainly would not caution against it. My parents always do the central park thing when they visit NYC, and they enjoy it. It really isn't my thing at all, so I don't recommend it on a personal level. But if the idea of a massive, pretty park in the middle of a massive, pretty city seems fun to you, then by all means go!
Not to be missed: The
High Line Park. This is my new place to bring out of towners. It's amazing. It's a park that goes up the length of the lower, west side of manhattan on an old, abandoned train track and it's \ impeccably designed. You could make a really fantastic afternoon taking a spin through the
High Line, Visiting the
9th avenue Apple Store (At 14th street), and then taking a jaunt through the
Chelsea Market. For food, grab a slice of pizza (no specific recommendations for where, in that area, that I can think of) and eat take it up to the high line. Lots of seating up there.
For what I consider a very authentic NYC experience, you have to go to a really cheap, poorly lit Chinese takeout place to get greasy, yummy, americanized chinese food. Yummy! No seating! Young Chow near Union Square is a great choice, and is your quintessential, NYC chinese takeout. If you're not into chinese food or friend stuff, pass this by

. I feel in heaven at these places though. Also, dollar pizza at 9th avenue and 41st street is fantastic. Greasy 1 dollar slices on the thinest crust imaginable, with cheap yummy cheese. Not fine dining, but certainly a fun NYC place to grab some grub.
Enjoy your trip,
Emerson