By road
Believe it or not, it is possible to hitchhike out of the New York Metro area. If you are trying to go long distances, your best bet is to take NJ Transit or Metro North far enough to put you well into the suburbs, preferably to a stop that puts you near i.e. within walking distance of a major highway such as an Interstate. From there, get to an on-ramp and put out your thumb. Be advised, however, that New Jersey state laws on hitchhiking are notoriously ambiguous, and you may be hassled by local police, so use common sense and discretion.
If you're trying to go west into Pennsylvania, a good tip is to take NJ transit to Mt. Olive, which is only a 5 minute walk from I-80, which generally carries a good amount of long-distance traffic going west.
By train
Amtrak (http://www.amtrak.com/ser...) operates a line the Northeast Corridor through NJ. It goes through NJ from Philadelphia to NY Penn Station to points beyond Boston in the north, and Washington, DC and Newport News, VA in the south.
SEPTA has service into Trenton from Philadelphia, and can be used in Southeastern New Jersey.
New Jersey Transit has rail service from New York City. NJ Transit rail service can be used for central Jersey and Northwest Jersey. One can also take the PATH train from NYC into New Jersey or PATCO from Philadelphia into New Jersey.