There are many trains that go through New York, especially through Pennsylvania Station in New York City.
From New England:
Acela Express: is a high-speed train connecting Boston, Providence, and New Haven to New York City.
Northeast Regional: is a train that also follows the same route but cheaper.
Lake Shore Limited: goes from Boston to Albany and Buffalo.
Ethan Allen Express: comes from Vermont to New York City via Albany.
Vermonter: is a train from Vermont to New York City via Springfield and Hartford.
From other states in the Mid-Atlantic:
Acela Express: is a high-speed train through New York City from Philadelphia, Baltimore and Washington, D.C.
Northeast Regional: is a train from Virginia that follows the same route as Acela Express but cheaper.
Keystone Express: comes from Harrisburg to New York City.
Pennsylvanian: goes from Pittsburgh to New York City.
From the Midwest:
Lake Shore Limited: goes from Chicago through Cleveland, Buffalo, and Albany to New York City.
Cardinal: comes from Chicago via Cincinnati and Washington, D.C.
From the South:
Carolinian: goes from North Carolina through Virginia to New York City.
Palmetto, Silver Meteor and Silver Star: go from Florida to New York City.
Crescent: goes from New Orleans to New York City via Atlanta.
From Canada:
Adirondack: goes from Montreal to New York City via Albany.
Maple Leaf: comes from Toronto through Buffalo and Albany to New York City.