I offer a few of the gems and pitfalls that I have discovered since moving to the city. Since I don't make very much money, I try to find cheap thrills. There are some great deals in city and many fun freebies (especially in the summer), but you may have to look... or check out a few of my recommendations.
Quick Tips:
Wear comfortable shoes. Most people (especially Americans) are ill-prepared for the amount of walking they will do in the city. Even when you use the subway, there are quite a few stairs you'll have to climb in/out of the station, plus final destinations are usually a few blocks away.
Go to the bathroom before you go out and plan your breaks carefully. There are very few clean, safe public restrooms in the city. Unless you want to get stuck buying a coffee at a Starbucks and standing in line for 20 minutes to go, make sure that you use the bathroom whenever you get the opportunity. Trust me on this one.
Best Way To Get Around:
The subway is easily the best way to get around. Go for the Unlimited Ride Metrocards. They will let you get on and off whenever you want, and you can use them on city buses, too. If you need to get across town (east to west), your best option might be a bus. Free bus and subway maps are available from any occupied subway booth (located outside most major turnstiles).
WARNING: Make sure that you want to enter before you swipe your card at the turnstiles. There is an 18-minute delay allowed between swipes. You can, however, get around this by walking up and catching a local bus - transfers to/from buses are free within 2 hours of a swipe.