Shoulda came a year ago when they still had the 1-day fun pass lol...that was the stupidest decision ever imo to cut the fun pass
But yea, go to a machine at the subway, get a metrocard. I recommend you pay the machine $37.85 and you will get 18 rides even with no leftover (they have some retarded 7% bonus now which basically will add a 7% onto your metrocard total and leaving you with odd amounts leftover like $1.81 which then you will have to bring pennies to refill at a token booth which most of the time they won't even do it for you - it's a long story lol). Or, if you plan on being here for a week, get a 7 day unlimited metrocard which will give you unlimited rides on subways and NYC Local bus, LI Bus and Westchester Bee Line buses for 7 days. (there is also a 30-day one if you plan on being here for a month) The weekly Metrocard is 29, and the monthly is 104. The minimum purchase for regular metrocards is 4.50; they will not sell you a metrocard with less than 4.50 with it.
The fare for the subway or bus is 2.25. With your 2.25, you can take one bus and one subway or two buses in any order. If you enter the subway or same bus route twice, you will be charged again. You can ride the subway as much or as long as you want for 2.25 as long as you stay within fare control (the turnstiles); once you exit you will be charged to get in again. All subway and bus or bus and bus transfers are valid within 2 hours and 18 minutes; you will be charged again once that time window passes.
To use a metrocard, as said before swipe it through the metrocard reader on the right hand side of the turnstile. If you did it correctly, it will tell you to proceed (GO), and if you have a regular metrocard it will show how much $$$ you have left on it. (if you have an unlimited it will just say GO).
If you ever visit Staten Island, swipe at St. George with a regular metrocard and you will be given an extra transfer to use on certain bus routes if going away from Manhattan or to the subway if going to manhattan.
PATH accepts Metrocards too, but since they are not run by the MTA they have a different policy. They do not accept any type of unlimited metrocard, they do not take or give transfers, and the fare for PATH is $1.75 which will be an additional charge to your metrocard regardless. When fanning PATH do not go past Journal Square unless you plan on paying again. I don't know whether Westchester Bee line takes unlimited cards or not as I have never done it myself.
That's pretty much all you need to know about the Metrocard in a nutshell (besides it's history and everything), if you're still confused there is always the MTA website to help you:
http://www.mta.info/fares/Good luck!