1st day: A tour in the city, we found one where you see almost everythink. Maybe catch a Broadway show that night.
2nd day: Visiting the places we enjoyed the most during our tour and didn't spend much time. Walking around manhatan (Greenwich, soho, tribeca, Central park, little italy, chiantown and harleem)
3rd day: Crossing Brooklyn bridge, Visiting some museums, and walk around in the city, specially the places we didn't had time to go in the day before.
It's easy enough to get to Manhattan from any of the three airports using public transportation - although easiest, in my opinion, from JFK. A cab from LGA or JFK can run anywhere from $35-$50, which isn't bad if there are a few of you and you want to get to where you're staying quickly. Another option is to reserve a ride with Super Shuttle - I think the last time I used it I paid something like $12.
A tour of the city is a fun idea, just to get "the lay of the land" so to speak. Regarding the Broadway show, you might want to keep in mind that to see a show and not pay full price (most places seats run for $110 a ticket now, I do believe, for Orchestra seats), many people stake out a claim at the TKTS ticket station in the middle of Times Square. The wait can last a very long time, cutting into your tour time. Check out specific theaters of the shows you're going to see, sometimes they have ticket lotteries a couple of hours before the show, although I'm not sure how many friends are in your group. Off-Broadway/Off-Off Broadway shows are just as good, and sometimes easier to get tickets from the theater.
To hit the Village, Central Park, Little Italy, SoHo, Tribeca, Harlem, and Chinatown in one day will kill you. You could probably cram Little Italy and Chinatown in in an hour or two (walking down Canal street is just fine, in my opinion!!), but Central Park is huge, and the weather will hopefully be nice enough by then that you'll want to spend a few hours there. It's worth having a meal in the SoHo/Tribeca area too, and if you plan on shopping, that will take some time as well; same goes for all of the little places in the Village, and the stores opening up in Harlem.
If you're going to walk across the Brooklyn Bridge, hang out in Brooklyn for the afternoon/evening! Many people don't make it across -- but there are some great things over there, such as the Williamsburg neighborhood and the Botanical Gardens, among others.
I would suggest mixing and matching by area . . . for example, hit the Met on the same day that you get to Central Park and Harlem (that's all uptown). Also -- plan on spending much more time in the museums than you think you will.
I've been in the city for years, so if I can help you out, I'd be happy to do so.