Buy a Metrocard. The last time I checked subway rides were 2 dollars one way but they go up periodically. I am old enough to remember when they were 1 dollar.
You need to know one of two things: Uptown or Downtown. The uptown 6 for example will take you in the opposite direction of the downtown 6!
If you need to get somewhere, know what street it is on. And the crosstreet. Not every train stops on every street, but if you can get close enough you can walk the rest. For streets like 42nd or other big streets, it's helpful if you know the crosstreets.
One train will not go to your destination sometimes and you will need to transfer. Transfers are free. You will need to know Uptown/Downtown. This is important. For example to get to 23rd street from Canarsie, Brooklyn, you'd take the L train to 14th Street Union Square, then transfer to the UPTOWN 6.
Get a train map. If you don't want to look like a tourist, they have them inside the train cars also. Don't be afraid to look over a New Yorker's head if they are sitting down. We are used to this by everyone, New Yorkers, tourists. etc.
Know if your train is a local or express! If it is local that means it will stop at every stop, express will not. You don't want to miss your stop. Some trains have both local and express trains. There are signs over the train stations but you can always ask someone.
If you are confused and the train map isn't helping, ask someone. Most New Yorkers know the subway system better than they know their own kids!! Ok, maybe not their kids, but at least their pets!