There are a number of ways to travel around in New York City. The lease expensive is taking advantage of the city's subway and bus system that snakes throughout all five boroughs. MTA Trip Planner is the absolutely best resource for finding your way around on the Subway or the bus system. If you'd rather not use public transportation, you might want to consider using one of the "hop on hop off" bus tours that circle around New York City such as New York Sightseeing - Gray Line Tours. Of course you can always take a taxi cab or car service, but be prepared for the significant expense.
If you're really adventurous, you can rent a car and chauffeur yourself around town, but truthfully traffic in this city can be a nightmare (particularly in Manhattan and during rush hours). So, if you're planning to remain in Manhattan during you visit, renting a car would be the absolute worse way to travel. Let's get some more depth detail about each of these modes of travel.
For brief instructions on how to ride the New York City Subway system take a look at the article - Riding the New York City Subway Made Easy
The New York City Transit serves Manhattan, Queens, Brooklyn, the Bronx, and - through MTA Staten Island Railway - Staten Island. The subway is not difficult to navigate. Be sure to get a copy of the subway map (Available at most subway stations) and utilize their web site to learn about any train route changes due to track work (they happen very frequently on the weekends) and for $2.75 a ride, you should have not problem getting around the city.
The New York City Bus System is even more extensive then the subway system, and covers all 5 boroughs. You can get a bus map at most subway stations and like the subway, costs $2.75 a ride.
One of the best ways to get around the city is by Taxi Cab. Yellow Cabs are plentiful in Manhattan and they are a great way to get around the city when you are unsure of how to get to your destination by subway or bus, or you are traveling late in the evening. For full fare information and tipping instructions, take a look at the New York City Taxi and Limousine Commissions Website.
If you really want to travel around the city in style, you can hire a car service to chauffeur you around town. There are several available, such as Carmel Car & Limo Service or Dial 7 Car & Limo Service. If the cost of hiring a car service fits into your travel budget, its a great way to travel around New York City hassle free.