@Neodymium I do like museums. I was also thinking a Broadway show if there is one during the day.
Here is one bit of lore that you will find useful - Broadway matinees are typically performed on Wednesday (and Saturday) and the Metropolitan Museum is closed on Wednesday, so that works out for you.
Do you know what neighbourhood you will be staying in?
There are walking tours for different interests - history, food, architecture, e.g. There are also stores for every taste - little boutiques, huge department stores.
You will be safe on the subway just because it's so crowded, but don't stand near the edge of the platform, and don't hesitate to move away from anyone who looks dodgy. The subway will get you where you want to go, but it's not a garden party.
One important thing to know is that September could have some cooler days, but might still be extremely hot and sticky.
There's really so much. Do some research on what's available according to your personal interests and you should have a great time!