While every state has its own traffic rules, most common situations are standardized, and you won't find much in the way of major differences.
You can consult the AAA Digest of Motor Laws for state-by-state summaries of traffic rules. Here are the pages for Washington DC and New York; you can lookup any other states you'll be driving through. One common difference between some states is the law on cell phone use while driving: you can see that the use of a hands-free device is required in both DC and New York and text messaging is prohibited. If you really want to study up, you can review the driver handbooks for DC and New York State.
One important rule, which you are simply expected to know, is that a right turn on a red light is prohibited in New York City unless there is a sign specifically allowing it. It is otherwise allowed elsewhere unless there's a sign specifically prohibiting it or a red arrow light.
Driving in New York City itself, particularly Manhattan, is generally not recommended for visitors. Besides its driving culture, parking will be a significant hassle and/or expense; you should read about the city's parking rules and determine where you plan to park before you go. This will usually be an expensive garage in areas tourists are likely to be interested in.
If your plan is simply to visit New York City as a tourist, as opposed to other parts of the state, I'd take the train (or bus, for a cheaper option) and scrap the idea of renting a car. A car will be a significant expense and hindrance for such a trip.
Oh, and if you find yourself driving through New Jersey, it's illegal to pump your own gas there (not that the police will come after you for it, but you'll be asked to stop because the station can, at least theoretically, be fined). Don't bother asking why; just let the attendant do it.