Spoiler: the subway is part of the experience
NYC is a big city and the public transportation can take you everywhere. The best way to explore the city is walking around and riding the subway. I suggest you to organize your itinerary in areas. Let's say that during your first day you are going to explore the Upper East Side - and you are statying in Midtown. So, use the Subway to get to the UES and then you explore the area by foot.
Every subway ride costs $2,75 - and you can buy an weekly unlimited pass for $32 - remember the Metrocard costs $1. You can also use your credit card or virtual wallet to pay the subway through the OMNY system, on the turnstyles. Just tap and go.
Google Maps is very reliable to plan your subway routes - just make sure to check the NYC Subway app on the weekends cause some lines and routes change.
Go check out my guide for the best free things to do as well as itineraries and travel tips to make your trip unforgettable.