One way you could make it more authentic is by staying at an airbnb vs. a hotel (airbnb's can be cheaper, just make sure you select one well-reviewed and not too in advance from your trip date) so you can live among locals. Wake up early on weekdays to get on the train during rush hour (between ~7-9 AM) you'll get a sense of all the different cultures and social classes represented in NYC. Go to a cozy coffee shop with your laptop or book (avoid Times Square, Herald Square...all the touristy squares lol) and just do something productive, you'll see plenty of local New Yorkers in coffee shops. I think Washington square park is really nice to hang out in after sundown, or you can go to a bar, etc. Also, do explore Brooklyn, Queens, and the Bronx - but be aware these boroughs tend to also have concentrated areas of tourists (eg. almost every tourist visiting Brooklyn will go to Dumbo).