New York has some of the best public transportation in the world-remarkable considering the subway system is over 100-years old-but to really get a sense of the city you need to see it on foot. The streets uptown are arranged in a grid pattern, but below 14th Street can be tricky, so you will need a map and the occasional assistance of others (don't be shy, New Yorkers like to help, mostly just to get you moving in the right direction rather than standing in their way). When you need to get somewhere a little faster, however, there are a lot of options.
Citibike, an urban bike-sharing service that has been in New York since 2013, offers two passes that will appeal to tourists: a 24-hour and a 7-day. You can ride as many times as you like during the length of your pass, and they will cost you 9.95USD and 23USD respectively. Remember that you'll be riding in a very fast-paced city with aggressive drivers, so stay away from congested areas and avoid rush hours. The city parks along the Hudson have many nice bike paths and cycling in the smaller downtown neighborhoods is a charming way to explore.
The subway, though grimy and crowded at times, is a quintessential New York experience, and is almost always the fastest way to get anywhere in the city. If you buy the 1USD Metrocard-a sturdy plastic pass that can be refilled as many times as necessary-the standard fare for one ride is 2.50USD, otherwise a single fare paper ticket is 2.75USD. Tourists may want to buy a 7-day unlimited pass for 30USD, but you'll need to be making more than 12 one-way trips to make it cost-effective.
The bus system is also a fine alternative, and will give you a chance to people-watch and see the city in motion, though it will be a little slower. They are especially useful when you want to get across town east-west, as the subway lines generally run north-south. The Metrocard can be used on all buses.
The yellow cab is another classic New York method of travel and, except during rainstorms or rush hour, they are very easy to hail. Just look at the light on the roof of the cab: if the center panel is lit and the sides are dark, then the taxi is available, if the whole panel is lit up then it is taken. All cabs should have working credit card machines installed in the back.