Addressing the all important question of...what is the best way to travel between Philadelphia and New York?
Cheapest is probably Megabus or Boltbus, especially with advance purchase. They'll take you from behind 30th St. Station (Amtrak). 2:10 if traffic behaves.
Slightly more expensive is Greyhound/Peter Pan, maybe $12-$15 depending. Takes you from the bus terminal to the Port Authority.
You can take Septa commuter rail to Trenton, and then switch to NJT to NYC. Costs about $25. Can get through tickets from ticket machines at 30th St. Station, or just use a Septa ticket and buy an NJT ticket at Trenton. 2.5 hours.
Amtrak. A walk-on ticket will cost you $53 most likely, except for peak of peak times, and advance will cost you $39. 1:20.