Dawg living in NYC here... agree on most everything here.
Food:
Pizza - Juliana's in DUMBO is the best in the city; it's at the foot of the Brooklyn Bridge and delicious. After eating, walk down to waterfront and you can see awesome views of downtown / Brooklyn Bridge. You can walk back across Brooklyn Bridge or take Subway back to manhattan.
On Manhattan, depends how you want to go... cheap and fast or more sit down.
A few places that haven't been mentioned before for sit down are Motorino in E. Village and Otto in W. Village.
NOTE: agree 100% with the poster who said NYC pizza is cheese or minimal toppings. I grew up in the south, but now nothing appeals to me more than a good slice of cheese with some basil
Burgers - Whitman's in E. village is good, but can be a wait. Have also been to Benson's down south of Houston ("How-ston... don't get me started...) that had delicious, massive burgers.
Italian - Can spread the gamut here, but I like smaller hole in the wall type places.
Patrizia's on 2nd avenue (~26th st) has good stuff and solid pizza, but can be loud.
Mimi's in midtown east is a hidden gem with a piano bar atmosphere
Activities:
Museums - 9/11 museum is somber, but very well done. It's close to One World Trade observatory, and Battery park so you can see the statue of liberty. If you have time and want to get close to the statue of liberty, you can take ferry to staten island for free, or pony up for a tour to go to the actual island. I think at this point, all of the tours to climb to the crown are taken until February.
Just went to the musuem of natural history (or "zoo without the poo" as I call it) with my parents. Interesting, but San Diego Zoo is cooler.
The Met is interesting if you like art and antiquities (recommend the American wing).
The Intrepid Air & Space is interesting with a bunch of military exhibits / old aircraft, including one of the space shuttles (that probably should have gone to Houston, TX rather than NYC, but I digress...)
Entertainment - If you're not offended easily, a comedy show is always enjoyable, although it's generally 18+ (small cover charge and a 2 drink minimum).
There are broadway shows that are funny and not too girly if you're not a theater nerd, although I haven't seen any shows lately so can't recommend any personally... ask at the TKTS booth what's playing and the folks can give you a run down on what each show is.
Sights - Central Park is really neat. I would recommend finding Belvedere Castle, as it is a really neat spot mid-park and has good views. You can work your way south from there and see the Bethesda fountain / Boat house and "the mall" which are the parts you'll recognize from lots of movies/TV shows.
If you're here before 1/6, the Rock Center tree is a must, but it's crowded. Christmas lights on 5th ave are great too.
Times square - see it once, then avoid it.
Transport:
To/From LGA - Might be cheaper to take the Q70 to the subway, but for your sanity, just pony up and take an Uber. If you're staying in Midtown manhattan, an Uber to midtown from LGA is ~$45, well worth the ease to get you into the city.
Around town - Subway is your friend. Generally, "Uptown" is going north, "downtown" is going south. Buy a metrocard, but unless you're here for a full week, probably best to just use "value" rather than "time" option.
Walking - you'll do a lot of it. Remind your wife and daughter as they make shoe choices that comfort far outstrips fashion when walking around a bunch...
You'll have a blast. If I see you around, I'll definitely give you a "Hail State".
Gotta disagree wholeheartedly with Juliana's in DUMBO. Both tourist trap pizza places in DUMBO actually. Those aren't even the only locations. I worked nearby for a sports agency and always got a kick out of people in their fanny packs lining up to eat there when Front Street Pizza is a few blocks away and infinitely better. The pizza in NYC is just good, just don't buy 'dollar slices'. Prince Street, Joe's.... don't wait somewhere in line.
Walk the Brooklyn Bridge, take a left and walk back down to 55 Water Street to the renovated factory that houses the West Elm HQ now. The roof is a public space with premium views of New York from 6 floors up.
The Cannibal near the Empire State building is a butcher shop that doubles as a restaurant. It's burger has won some pretty impressive awards. Their tapas are good too.
Little Frankie's in the East Village/LES is amazing Italian but cash only.
There's a... flea market? but nice... not sure what to appropriately describe it... called Chelsea Market I'm sure your family would enjoy. It has great food.
If you get down closer to Downtown Brooklyn you should visit their museum and garden. It's one of the nicer ones in NYC and next door to the Grand Army Plaza and the entrance to Prospect Park, which was designed by the same person who made Central Park.
If you're willing to wait in line for pizza, the only thing I'd actually recommend waiting in line for in NYC is Sugarfish. Premium sushi that doesn't take reservations or tips, and there's generally a line except between 1-5pm if you do a big breakfast and late lunch. Their 3-course meal is $23 per person inclusive of tax and tip. Hands down the best sushi in America.
The Central Park Zoo is nice and small and cheap. It's next door to the arsenal that housed weapons during the war, and that area is generally the nicest part of Central Park to walk around (pond, horse carriages, Plaza Hotel, etc.). You can actually walk from there to the boat house, i'm not sure about the availability in the winter but they have little jon boats you can rent for $15 for an hour and paddle around. It's nice.
Bottom line. Plenty to do and see. Very expensive. Be prepared for almost knocking out a $100 bill for breakfast at Juniors. Get same thing at Starkville Cafe except cheesecake.
Going to NYC and eating at Junior's would be like me going to NOLA and eating at Margaritaville (or worse, me going to NYC and eating at Bubba Gump or Hard Rock Cafe). It's overpriced because it's for tourists who hang out in Times Square, and the food is basically the quality of the pre-made stuff you'd buy in the back of Kroger. Find a nondescript diner and it's 1/2 the cost. In fact, outside of my rent, I don't think NYC is
drastically more expensive than Dallas or Nashville if you avoid tourist traps or hotel restaurants.
You'll have fun, you'll probably go the wrong way on the subway at least once, someone will probably tell you to GTF out of the way, but it'll be awesome and I hope you enjoy it. Just know that your favorite parts will probably be things you stumble upon and didn't plan for. That's primarily the case.