OK....as a former NY'er whose now takes my girls to NYC at least 5-6 times a year (DH still works/telecommutes to the city), here is what we do:
Again and again-
Children's Museum of Manhattan
Swiss Cottage Puppet Theater (in Central Park)
Children's Museum of Arts on West Lafayette Street (wear old clothes)
Natural History Museum (dinosaurs and planetarium)
China Town/Little Italy- Just go to pick up bakery items from both sides of Canal Street (Tai Pan Bakery for roast pork buns and Ferrara for italian cookies and cannoli)
FAO Schwartz
Toys R Us
Macy's
Ollie's Noodle Shop
Magnolia Bakery (in the Village)
Cupcake Cafe (on East Side)
Central Park Zoo
Some things we've done just once or twice (but dd liked):
American Girl Tea- We loved this and it was only 20.00 a child (but spent much more after cruising through the store...guard your charge cards here!). It's cute...it's festive! We met one of her friends here when we discovered we would be in town at the same time.
Staten Island Ferry- So Papa could relive his boyhood....fun, but cold!
Subway Museum (first stop on the 2 into Brooklyn)- I used to take my first graders here when I taught in NYC. They can climb on all kinds of old subway cars!
Met Museum Shop at Rockefeller Center- Go upstairs for great art related kid's games/toys. Santa shopped here a lot of dd4 last year!
Lower Eastside Tenement Museum- May not fit the mood of your trip, but they have great dress-up opportunities for the kids and, if NYC immigrant history interests you, it is a very, very interesting hands-on museum. You actually tour a tenement set up in three different time-periods where they tell the story of three-different families. Again, may not fit the Broadway Play and Shopping mood- but put in on the agenda for some trip in the future. Definitely a hidden gem!
Meals-
Here's my best advice- You don't have to spend a lot to get a good meal in NYC! If you are in the theater district, go west to 9th and 10th Avenues and skip the theater menues...great restaurants....not so touristy...but very safe!
Carmine's is good...but a bit touristy (although a lot of locals eat here too) and a long, long wait! However, I also think Little Italy restaurants are touristy. We like Angels on the upper east side for good Italian and DH swears by Patsy's for pizza. I would sent you to Arthur Avenue in the Bronx or Aunt Suzie's in Park Slope, Brooklyn...but you won't have time. Save these for another trip...they are worth the trek!
I like City Grille on 72nd and Columbus.
Our "go every time" is Delta Grille on 9th avenue (just west of the theater district).
DD likes Tom's Diner on the 112th/Broadway...not fancy, but great food. It's the diner whose exterior was featured as Monk's in Seinfeld. Great egg creams....and open all night- logged some early morning hours here in graduate school.
Deli- We love the matzoh ball soup at Artie's Deli at around 72nd and Broadway. We often go here after the Children's Museum or a trip to Central Park.
Spice on the Upper East Side has great Thai.....but the Lemon Grass (a chain) does just fine. You can find the Lemon Grass in the Village and on the Upper West Side.
For fun:
Serendipity- frozen hot chocolates
Dylan's Candy Bar- ice cream
Peanut Butter and Jelly Factory- (a fun little hole in the wall near the Village)
Magnolia Bakery (for cupcakes...dd just likes the frosting)
Central Park- We also just like to hang out at the playground on 72nd and Central Park West.
I'm really interested in this thread since we visit so often and it is usually just me with my two dd's all day while DH is working. I think we've turned NYC upside down taking advantage of every kid-friendly activity! I've learned to never pay for an expensive meal (unless I really want to try a special place). Also, we take the subway....families who live in NYC do this....why shouldn't we? DD loves it....but I do take cabs when I am pressed for time or just sick of lugging the stroller and two girls down the steps!