A week should be perfect for you and the family to enjoy the city. I find people who only stay 3-4 days try to pack too much in; always better to go slow and take things in rather than rush around trying to fill an itinerary.
For sightseeing, the double decker buses are a nice way to knock out the major stops. I always strongly recommend Ellis Island over Statue of Liberty - both can be done on the same ferry, but linger at the former. Related - it's a bit out of the way, but the Lower East Side Tenement Museum is an amazing experience. Both will be highlights of your trip.
There are of course several amazing art museums that are worthwhile - Guggenheim, Metropolitan, and MoMA are my personal favorites - but for 10 and under nothing beats American Museum of Natural History. Also, the balloon inflation is on the four blocks surrounding AMNH (Central Park West, W 77th St, Columbus Ave, W 81st St.) If you like smaller museums, the Frick Collection is amazing. Neue Galerie (near the Met) should be on your list as well - relatively new, specializes in German and Austrian art. You should also check out the Whitney Museum of American Art, which is re-locating to the Meatpacking District May 1. It will be right next to the High Line, which is a must see as well.
Restaurants - although I still love Zagat's guides for both dining and shopping, Yelp is an incredible online resource for planning. Can be tough doing meal planning on the go. NYC is relatively expensive but the dining options do run the full spectrum - you can find affordable places to eat if you plan it out.
Other ideas:
Staten Island - Yeah, no. There are interesting things. Chinese Scholar's Garden (sculptures) at Snug Harbor. Richmondtown is a wonderfully preserved period village. I dated a S.I. girl for six years and know the island deeply. But just no. You can take a roundtrip on the Staten Island Ferry to kill an hour, get a NY Harbor perspective, and it's free.
Queens - Again, many interesting sites, though spread out. Science museum, Queens Museum of Art has an amazing model of the entire city, Corona Park and the 1964 World's Fair Pavilion - both near the Billie Jean KIng USTA complex and Citifield - are all cool day trips, but imminently missable.
Bronx - The New York Botanical Garden is pretty amazing. Not sure when the Holiday Train show begins, if it's that week you should go. Great Italian food along Arthur Avenue right outside NYBC. Bronx Zoo is nearby. It'll be a little cold to take in Wave Hill or Pelham Park. The Cloisters is a pretty cool museum with pretty neat tapestry and amazing vistas. But all of these worthwhile ideas are close to a full day trip, so you have to decide if your family would enjoy it.
Brooklyn - Oh, hell yes. This was my home for eight years and I would love to be your guide for a day on this excursion. Walk the Brooklyn Bridge, eat in DUMBO or Brooklyn Heights. New parks with many activities along the East River. BMA has some great collections, Prospect Park is comparable to Central Park, Brooklyn Academy of Music is the finest avant garde venue in the city. I could write a book on things to do.
Chinatown - plan on half day to stroll through, do some shopping, maybe grab some dim sum. Not far from Lower East Side Tenement Museum.
SHOPPING - SoHo is a nice half day, as would be Fifth Ave between 42nd and 60th. These are both presuming you're prepared to drop some serious coin.
OK, so for the sightseeing stuff, they have these double decker buses they imported from London. As far as I know it's one company so it shouldn't be hard to find online. Not sure if it makes sense to purchase in advance - there's no issue with availability but might be a discount? 1/2 day, 1 day, 2 days - whatever you do, that takes care of the major touristy stuff (from north to south: Central Park, Rockafeller Center, ESB, Macy's, Greenwich Village/Washington Sq Prk, Wall St, et al).
So, figure 1-2 days of sightseeing.
You asked about visiting the UN - tight security AFAIK and not much in the way of touristy stuff. I have never had guests visit. Columbia University is far west side, could be combined with a visit to Grants Tomb. It's an urban campus without any real stand alone attractions. Same for NYU, which we think of as the Purple Blob which ate Greenwich Village. Many other Universities but those are the most prestigious. New York is very dense and there are no park like campuses in Manhattan.
Shows - lets table for now. Matilda is a great show for kids 7 and up. The Lion King is a classic. Many long running shows & revivals if you have one in mind (Chicago, Once, et al). I have some good online resources for discount tickets, and tips/tricks on how to score last minute deals. A lot will depend on what's running - we'll have a better sense what to shoot for in the fall after the new season begins. I live in a theater building and have good connections for off-Broadway and Off-off Broadway.
Accommodations - I live in Hell's Kitchen, which is midtown west & borders the theater district. Where to stay is a major decision & I'm not sure how much help I'll be. Yotel is new/near and we have many budget - its a relative term - hotels, both old and new, nearby. Times Square or near TS will be more expensive. If you find one that fits your budget let me know and I'll give you the lowdown on convenience factor. Also, Google the bedbug report - tells you if the hotel has had violations. I joke a lot but I'm being serious here.
Mass Transit - week long metro cards for both buses & subways (5 x $30, everyone taller than 44" needs one.) General rule: north - south take trains, east - west hoof it or take buses. Taxis are more expensive AND take longer than subways. Citibike share is awesome but probably not an option with your youngest (is he four?)
The Grid - 200 years ago they laid out the plan and north of 14th it couldn't be easier. City blocks run east-west and take 1 minute to walk between (going north-south); Avenues run north-south and take 3-4 minutes to walk between (going east-west). Everything is numbered - just keep your eye on the Freedom Tower or ESB to stay oriented. The old Kings Highway - which the Continental Army scampered up after getting their asses kicked in the August, 1776 Battle of Long Island (Brooklyn) - cuts diagonal across the grid from The Battery all the way up to the Upper West Side. Today it's called Broadway. Below 14th the streets are still a grid, but the Financial District and the Village don't always line up because they were separate plans which predate the master plan.
Some suggestions on itineraries...
There are issues with availability and wait times for Ellis Island/Statue of Liberty and the 9/11 Memorial. Buy both in advance online to reserve your spot and avoid a lengthy queue. Those are both downtown along with Wall St. That's a full day for those 3 things.
Walking the Brooklyn Bridge & exploring that borough is another day. Maybe this is the Sunday after you arrive.
Chinatown could be combined with the Lower East Side Tenement Museum. That could also be combined with exploring Soho & GV/West Village OR The Whitney Museum + The High Line + galleries of Chelsea. This is old low rise New York, mostly 6 story buildings. This is MY New York, the one that seduced me 16 years ago and never let go. A 1-2 hour walking tour is a lot of fun - I can recommend a guide.
Balloon inflation of the parade is the night before Thanksgiving, Macy's parade is the next morning, turkey & traditional meal in the afternoon. So this is another day. AMNH could be a full day or a good chunk of it - that could be your Wednesday.
That suggested outline is 5-6 days already so I'll stop now.
We have lots of time to plan. Are you guys planners? Or loosey-goosey go with the flow, let's figure it out after we land travelers? You're German! Of course you plan.
I think, in general, loose itinerary planning with lots of time padded in works well. It's nice to have some time available for spontaneity or to allow yourself to linger if you are enjoying something. New York is not a city but rather scores of neighborhoods. Each has its own characteristics and ethnic flavor with unique stores and restaurants. Even people who live here love to dedicate a day to just learning & exploring a new neighborhood.
Plus, kids. They're the best thing in life, right? But #### do they ever screw up the best of intentions. So everything will take a little longer than you figure. No sense stressing about it, just go with the flow.
Airports - no preference as they all suck equally and take roughly the same amount of time to midtown. Gun to my head, EWR > JFK > LGA.
General geography:
Manhattan is the city. The other four are outter boroughs (Bronx, Brookkyn, Queens, Staten Island). Big sweeping generalities: Brooklyn is cool, the other three wish they were. White people are blithely unaware of anything above 96th St.
Uptown - anything north of 59th St. Two general classifications are Upper West Side (progressive, liberal, Woody Allen) and Upper East Side (conservative, Republican, Bernie Madoff).
Midtown - 34th St to 59th St. People work here. Tourists overrun this. Neighborhoods like Hells Kitchen on the west or Murray Hill on the east but generally its tall buildings and commercial ventures.
14th St to 34th St - Chelsea on the west, Madison Park in the middle, Gramercy/Kips Bay in the east.
Below 14th Street - downtown. This is old New York. The Battery is the tip. The Finacial District and Battery Park City are above that. Continuing north, TriBeCa, City Hall & Chinatown are next. From left to right above that, Hudson Sq/West Village, Greenwich Village/Soho, Lower East Side/East Village. This is the soul of New York; if there is anything authentic left in the ever changing and ever evolving city, you'll find it here.
Some other landmarks to orient yourself to:
Central Park runs from 59th St to 110th St, from 5th Ave to 8th Ave. ESB is 34th St/5th Ave. The Freedom Tower is all the way downtown. None of these are easily combined destinations.
(Also, Columbia U is FAR UPPER west side - 116th St - and Grants Tomb is 122nd St.)
American Museum of Natural History (AMNH) - Central Park West, W 77th St, Columbus Ave, W 81st St. It's the whole block.