It's possibly easier to narrow down advice based on what you and your family like to do rather than just rattle off all kinds of things. March was gorgeous here this year, the sun was out every day and it was pretty warm, so it's a hard place to pack for as well! I'll mention a few things, but feel free to ask for more info or reach out privately about anything.
You'll be surprised how much ground you can cover on foot as well, I'd recommend walking a lot rather than taking cabs and subways unless you're going a significant distance, you never know what you're going to see along the way. Although having said that, the subway is a wonderful service.
I'm not a fan of Times Square, but you can't come all the way and not take a walk through it... although to be honest it's little more than a brightly lit area IMO.
March might still be a bit early for some things, Coney Island is still in bad shape from the hurricane, for example, as is Staten Island (as well as Long Beach and many other places by the water), although the ferry is running and highly recommended as Neil says. You do little more than get the ferry to Staten Island, got off and hop on another one and come back, but you get some great views of lower Manhattan and it goes fairly close to the statue of liberty (I'd recommend heading out to the back of the ferry on the return journey and watching the city get closer). I think the statue itself is still closed to the public for repairs but I could be wrong. Another easy/free thing to do is to take a walk along the Brooklyn Bridge (if you can time it for dusk then even better), better to walk from Brooklyn to Manhattan.
A handy tip for both The Met
http://www.metmuseum.org/ and the Natural History Museum
http://www.amnh.org/, both are donation-based so you can pay what you want rather than the "suggested" fee (which gives little indication that it's merely a suggested donation). Both places are huge (especially the Met) and are really all-day activities although if you don't have that kind of time to spare you should at least pop in for a bit to get a feel of the place. You have to pay extra for a few things in the natural history museum (like the planetarium) but paying a few bucks to get into the Met gives you access to almost everything. Not sure the roof will be open in March, I think it opens in May (but if it is open you can get up there for free).
You'll need to walk around Central Park for a bit, of course. Many people like to go to the Strawberry Fields memorial for John Lennon, and just outside that part of the park is the Dakota building where he was shot...
The Bronx Zoo is a great day out, it'll be open from 10:00am-4:30pm during March and you'll need every minute of it. Another tip is that on Wednesday you can pay what you like to get in.
I've yet to do some of the things I'm sure are great (Empire State Building, Top of the Rock, MOMA, Guggenheim). I also hear that the Tenament museum is fantastic, am planning a visit soon.
There are loads of great inexpensive places to eat as well, the first thing I think about when thinking about recommendations is food! Beer is expensive though although since it's a family holiday you might not have to worry too much about that. If you do go to a bar, worth remembering that the etiquette is to leave a $1 tip on the bar per drink you've ordered (I wouldn't tip for a coke/soft drink though, just alcohol), it's fairly grating but you might not get served again if you don't... although that's if you're ordering at the bar rather than a table service.
Best coffee I've had:
http://abraconyc.com/Tiny little hole in the wall in the East Village. Get the drip-coffee.
Bakery:
http://littlecupcakebakeshop.com/locations.html (prince street), whenever we are in this part of town we have to go in. The wife thinks the pistachio cake is the best thing she has ever eaten.
Pancakes:
http://www.clintonstreetbaking.com/ (just be warned this place gets very busy)
Grab a falafel sandwich for $3 from
http://mamouns.com/, it'll keep you going!
The best burgers I've ever had are in our neighbourhood in Brooklyn from Ox Cart Tavern (
http://www.oxcarttavern.com/).
Two record shops to visit:
Other Music:
http://www.othermusic.com/ (then from there you can walk to Abraco, grab a coffee and go to) Kim's:
http://www.mondokims.com/ and if you're so inclined around the corner from Kim's is the building used for the Physical Graffiti album cover at 96-98 St Marks Place.
If you're looking to go to any gigs, keep an eye on
www.ohmyrockness.com (though you'll need to make sure a show is all-ages if you're taking everyone, most shows here are 21+).
