My first tip has to be museums. Why? Well where else can you spend an unlimited amount of time while learning new things and be warm when it's freezing or raining outside? I actually love a gloomy rainy day spent browsing a great exhibition. Oh and you know the "new" Whitney museum is opening in spring 2015 in downtown so we won't even have to travel that far for this one any more… My number one though in New York is the Met with their permanent collection of japanese warriors on display and pretty innovative fashion exhibitions.
David Zwirner gallery
Gagosian New York
Another way how to get cultured is to spend an afternoon walking through the Chelsea district. The Chelsea galleries are for free and open 5-6 days a week. Avoid coming on Sundays because you'll find closed doors and some of the galleries are also closed on Mondays. The majority shows pieces from contemporary artists but you might occasionally stumble upon modern art as well. The only drawback might be the fact that if there's an artist that is on high demand in New York you might have to stand in a pretty long line to get into the gallery. (Like when I was waiting over 3hrs in the crazy cold last winter to see Yayoi Kusama's Infinity room…)
Shopping. It sounds like it's the obvious thing to do but trust me it is so much better to shop during the fall or winter. I feel like the stores carry a much better selection compared to the summer. I was always able to find some amazing deals on originally sky high priced items… Look for Black Friday, Christmas deals and post-Christmas markdowns as well as the less obvious but a lot more exciting sample sales. Check out Racked or Stylish City for the list of current and upcoming sample sales. When there are none there's always the shoe department at Bergdorfs though…
And last but not least try the Citibike! I'm more likely to enjoy it when it's not too hot outside and hey who doesn't need to burn the calories from last night's margaritas...