Eat a gyro from a street vendor. My wife still swears it was the best food she’s eaten in New York.
Many NYC street vendors are very good. Roasted chestnuts are good in winter. Sabrett’s hot dogs are an inexpensive quick snack (gotta be a Sabrett’s - with the yellow and blue umbrella). There are many others but the next time I’m there I’ll want to try a gyro.
And, the NYC deli sandwiches are good, too. They’re on just about every other corner.
And NYC cheesecakes, yumm…
More importantly, you can buy tickets to matinees a day in advance at the Seaport. So if you are going all-out broadway, a viable strategy is to hit up a Tuesday Night performance and a Wednesday matinee (or a Friday Night and Saturday matinee) with a single trip to the TKTS booth at the Seaport. It’s relatively close to the 9/11 museum so if you are going to do that you can loop that in too. The line isn’t just shorter at the seaport… it’s MUCH shorter, like nonexistent shorter.
If your trip is before the Tony Awards in June, you should have good luck in getting into new shows not named Hamilton.
I wish this were still the case. I find that deli, in addition to a decent bagel, is a rapidly vanishing food type here.
If you want Starbucks, however, you can find those on every block in many neighborhoods. 
This would be my City Island day.
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Go to the Bartow-Pell Museum, which is not on City Island but is near City Island. It’s small (for a museum) but provides a good overview of the history of the area, and even if touring historic homes is not your thing, the grounds are really pleasant and interesting – beautiful formal gardens overlooking the water, a lot of natural areas with paths like parkland, an old burial ground, a carriage house, and information about the original Native Americans who lived in the area.
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Visit the City Island Nautical Museum, which, as a museum, is maybe not the greatest or most rigorous thing ever … but I think it is really cute and charming as a way of seeing how a small community highlights its own history. It’s in the former island schoolhouse, which is also extremely adorable. This could be a 20 minute stop, at the most.
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Eating. The thing City Island is most famous for is not really my thing, but I will tell you they have several casual seafood places, like clam shack kind of places, where a lot of things are fried, and plenty of people enjoy them. My personal choice of restaurant is The Black Whale, which has a more general American standard menu, and a great outdoor seating area. But there are tons of more seafood-specific places you could choose from.
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Walking around. Probably the best thing to do on City Island is walk around, just up and down the residential streets (City Island only has one main street, and the rest are residential). You will see adorable cottages, beautiful historic homes that have been meticulously kept up, ancient houses that look like they are from a Scooby Doo mystery, a few modern homes that you can debate how well they blend into their environment. You can also walk around Pelham Cemetery, which is right on a bluff over the water. On that same side of the island, you can see Hart Island across the sound, and the ruins of the old sanatorium buildings on it.
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Before, or during, your walking around, you can get ice cream from a little stand called Lickety Split, which is right on the main drag. If drinks are more your thing, The Snug is a very cute bar with a very good burger/sandwich menu, and it’s always really fun to drop in. Fella’s is a real dive bar with a pool table, also a good time. The Alehouse has the large beer selection and is perfectly fine (although the Snug and Fella’s feel more like local places to me).
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Especially if you are driving, or if you have a great interest in maritime history, you can combine your City Island day with a visit to Fort Schuyler, which is on the campus of the SUNY Maritime College. It’s a historic fort, and it houses the Maritime Industry Museum, which is probably the kind of thing most people have never wished they could go to a museum about. The grounds are impressive, you can breeze through the museum and mostly soak in the ambiance of shipping and being in an old fort.
This entire plan is really contingent on having good weather, because so much of it involves wandering around out-of-doors at your own pace.
From my daughter: broadwayforbrokepeople.com will tell you you what shows have rushes (and which ones are open only to those with student IDs), and which ones do lotteries.
Probably worthwhile to join the Hamilton lottery online, and if you’re there on a Wednesday, go to the Richard Rodgers Theatre at about noon for the in-person lottery. Before the 2:00 matinee, there’s (usually) a pre-show performance outside that’s free for everybody (kind of like buskers, except they don’t shove a donation bucket in your face).
Museums: The Met (81st St. and 5th Ave) is donation-only, so whatever you feel you can spare. MoMA (53rd St. and 6th Ave.) is free on Fridays from 5-8 p.m.
Sightseeing: If you feel like going up to the top of a building to look at the skyline, The Top of the Rock has the best view, as you can actually see the Empire State (and it’s the same price as Empire State $32). If you’re hoping to add the Statue of Liberty to your trip, take the Staten Island Ferry. It’s free and gives you just about the same views as the Ellis Island Ferry. And you can’t go wrong with a walk across the Brooklyn Bridge and a walk around Brooklyn Bridge Park–best panoramic views of Manhattan. As for Central Park, Strawberry Fields (adjacent to the Dakota, a dedication to John Lennon) and Bethesda Fountain are some nice spots. Other than that, just take your time and take it all in. Walk from one end to the other if you feel so inclined. There really are no bad views.
Food: Don’t be afraid of street meat. Best $6 meal you’ll have all week. If you’re in midtown east (51st and 3rd) stop into Ess-a-Bagel and grab an authentic NY bagel done well.