Justin
From Penn Station in NYC you have many ways to connect to trails
From Penn proper
Amtrak's Vermonter will take you to
Brattleboro where you can hitchhike and easy and scenic 60 miles across Rt 9 and pick up the AT at Bennington VT
White River Junction where you can pick up the AT just before you get to Hanover NH
Waterbury VT where you can take a cab to the Long Trail crossing
From Penn you can also take New Jersey Transit to pick up the AT at Port Jervis NY
Just up the block from Penn is the largest bus terminal in America ; the Port Authority Bus Terminal where you can connect to buses that go everywhere
including Martz Transit that will get you out to the Delaware Water Gap in about 90 minutes
and you can also take bus from the PABT to Bear Mountain
About a 1.5 miles from Penn is Grand Central Station where the commuter rail road Metro North runs out of
Metro North is good for AT stops that are in NY State and east of the Hudson or along the NY/CT state line >see joshuasdad's map
The stopped labeled Appalachian Trail is only used on weekends and in season : use the Pawling stop at all other times
and then there is Amtrak to Boston where you can take a Concord Coach Lines bus to interesting places like Franconia and Pinkham Notch