Good advice above. I’m old enough to remember Yankees bleacher tix being $1.50–we were incensed when they doubled them to $3!
Totally agree on the downtown advice—I’ve been an uptown/UWS guy most of my life (preference-wise; I’ve lived in the ‘burbs for ages), but only recently discovered downtown (thanks, kids!) Great bookstores, restaurants/bars/cafes, shops, vintage movie theaters, Washington Square park, etc. I go in and hang there most weekends. And there’s a Wegmans on Astor Place.
Off the beaten path, non-sports suggestions—
Central Park—just walking around from any entrance.
Museums—world-class. Met (most variety), Guggenheim/MoMA/Whitney—all more modern, Frick House—older stuff. My favorite part of living here.
Yakitori Totto—55th and 8th. Great Japanese izikaya bar food. Meat skewers. Great stuff. Not cheap, but not super expensive either. The kind of place you don’t find often outside Japan.
Broadway/Off-Broadway—world class theater.
You can do a month visiting here and still not see it all—enjoy!