To get to downtown Manhattan, like Greenwich Village, from the LIE take the Brooklyn-Queens Expressway, BQE, to the Williamsburg, Manhattan, or Brooklyn Bridges. To get to Columbia Univ. Be careful of parking on the street. Its free, but the parking regulations are byzantine, and often very poorly marked. Also, there are tow services which operate like bandits.
Parking lots can be expensive.
You can often do well by going to the extreme west or east in Manhattan, and then looking for a lot. A little patience can be rewarding. Parking lots run by the City are generally ok. Not covered, though. Also, taxis are relatively cheap and relatively honest.