We have covered this one before
One option (don't know if it's actually viable) would be to see a New York Yankees game at Yankee Stadium. It would depend on if they're playing at home when you go and if you can get tickets There is also the New York Mets. Located at subway 7; subwaystop: Mets - Willets Point Station. Also...
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The Subway museum is in Brooklyn, just across the East River from Manhattan, and if you are "allowed"

just one train jaunt, that's what I would do. They have it seems one of every old car type, and being rail connected can occasionally get them out for a run.
The Subway bookshop etc is inside Grand Central, and if you do the tour there (although it's as easy to just walk round and look) you should spot it.
A ride out from Penn Station to Jamaica (above) on the LIRR gives a good view of the substantial Sunnyside yards, used by several of the main line operators.
Best subway trip is to Coney Island, end of several lines, choose one which runs on the old Elevated viaducts. There is an enormous Subway depot and works in the last couple of stations before the terminus as the lines converge, surely the largest in the world. There are still quite a number of Elevateds on the Subway in the inner suburbs. At Coney Island have a traditional hot dog and beer ("dog and suds"), walk on the beach, ride the classic roller-coaster.
The Staten Island ferry is a far better ride than the Statue of Liberty, gives a close view of it anyway, and avoids the tedious queues on Liberty Island.
The most comprehensive site about the Subway is this one: