Not much to add to what has already been posted. Everyone has their favorite pizza spot, section of Central Park and priorities on how much time to spend in Times Square. For a first timer, it's pretty overwhelming to take it all in. Normally I would tell you not to stay only in the city and to venture out, but kids change that dynamic. So I personally would start at the bottom and work my way uptown. (I.E. SOL, 9/11 Memorial, Trinity Church, NYSE, Presidential memorial, Fraunces Tavern on Day 1; Meatpacking District, SoHo, LITA, Chinatown, Brooklyn Bridge on Day 2; etc.) One place that often gets overlooked is the Columbia/Inwood area where Grant's Tomb and Tom's Diner (Actually, restaurant) from the song and Seinfeld are.
Maybe I'm just used to it by now, but the food cost thing is a bit exaggerated. If you are going to do a foodie tour, yes, you will pay out the ass. If you are grabbing a few slices of pizza, some decent Chinese and a family dinner, it's not too bad. The only recommendation I will give you is to do Tony DiNapoli's with the family for your TS visit. (Make a reservation on OpenTable, as will need to do with most places)
You will need to take the subway, but the bus tours are great to get the lay of the land. Yes, you will get lost on the subway. But, it's like driving a stick: It takes a day of practice. When it comes to giving directions, NYers rival Texans in helpfulness. I recommend the app MdMost posted about as you will most likely not have internet service while on the trains. If you're doing the Met's game, get Long Island Railroad (LIRR) tickets rather than taking the subway. Less of a shit show, and actual train with seats and worth the extra $ to get there quicker. These leave from Penn Station.
Never heard of the Skyline Hotel, but I used to live in Hell's Kitchen. If you're a Seinfeld fan, the original Soup (Nazi) Kitchen and Studio 54 are in the area once you've seen TS.