We are in the middle of southern Connecticut, so we always just go over the Tappan Zee bridge (which is NEW and mostly finished now!) and loop around New York City and Newark. That is I287, exit 21 I believe in New York off of I95 (just after the Connecticut border). For us, this does mean going down I95 in Connecticut - which I have decided actually has worse traffic than New York or New Jersey.
As to the new Tappan Zee Bridge, stay out of the far right lane. I don't know if they will change them eventually, but they have water drains there that will shake and move you all over the place. Typical..... brand new bridge, and they have to put in something to remind you of how bad the old bridge was!