New York City is made up of five boroughs: Manhattan, Queens, Brooklyn, the Bronx and Staten Island. Manhattan is most likely where you'll spend the majority of your time. The other boroughs require you to cross a river to get there. Most of the popular hotels, theaters, museums, etc. are on the island of Manhattan and depending on how many times you've visited before, you may or may not choose to venture out to the surrounding boroughs. I guess you'll literally cross that bridge when you come to it.
There's the Long Island Railroad, the New Jersey Transit and the MetroNorth, but you will be dealing with what's called the MTA (Metropolitan Transit Authority) which consists of the New York City Bus and Subway System. When people tell you to take "the subway" or to take "the train", they are referring to the MTA subway system--any trains that run underground in Manhattan. So don't hop on that train in Grand Central Station because the cool guy on the street said to "take the train." Lovely as it is, I'd hate for you to end up in Connecticut.
Sure, there are plenty of parks with public restrooms and some subway stations but I would avoid those at all costs quite honestly, because what awaits you in those restrooms is always a mystery--a less than pleasant mystery. You're much better off strolling into a hotel lobby nonchalantly and making your way to the restrooms there. Also, some of the larger department stores or chain stores (Bloomingdale's, Old Navy, Apple, etc.) provide a pleasant environment for relief when nature calls. Starbuck's is a good idea in theory, but unless you're willing to wait in a 20-minute latte line, I'd steer clear of that option as well.
I understand that you don't want to look like a GAP window display gone terribly wrong, but no matter what the season in NYC, you'll be sorry if you don't layer. When you're out and about in 20-degree weather and then head into a building that's pushing 88 degrees, you'll thank me. On the flip side, when it's a humid, sticky 90, and you decide to go to the museum, if you don't have a top layer on-hand, you may very well end up frozen and put on display next to the wooly mammoth.
New York City is mapped out as a grid with streets running east to west and avenues running north to south for the most part. No matter what your form of transportation (Uber, taxi, walking, etc.), when asking for directions or looking at a map always ask for or take note of the cross streets. It will save you a great deal of time, not to mention backlash from upset New Yorkers as you stand dazed and confused in the middle of the sidewalk staring blankly at Google maps on your smartphone.