The simplest way would be to take the Downtown A-C-E (the blue line) from 34th Street-Penn Station to Canal Street.
You will exit the subway at the north end of the Tribeca neighborhood, just south of SoHo, at the intersection of Canal and Varick Streets. Walk one block north on Varick to Grand Street.
If you're going further east on Grand Street, then take the Uptown A or C (not the E) to 59th Street-Columbus Circle. Then transfer at 59th Street-Columbus Circle to the Downtown B or D trains (of the B-D-F-M, the orange line). Take the Downtown B or D to Grand Street (at Chrystie Street), which is in Chinatown.
Take the Uptown B or D trains (of the B-D-F-M, the orange line) from Grand Street (at Chrystie Street) to West 4th Street. Transfer at West 4th Street to the Uptown A-C-E (the blue line) to 34th Street-Penn Station.
If you're at the west end of Grand Street (near 6th Avenue), then walk west to 6th Avenue, turn left, and walk 1 block south on 6th Avenue to Canal Street. You can take the Uptown A-C-Estraight from Canal Street (at 6th Avenue) to 34th Street-Penn Station.
There is no train that runs directly between Penn Station and Grand Central. You have to take the subway: the Uptown 1-2-3(the red line), 1 stop, from 34th Street-Penn Station to 42nd Street-Times Square, then transfer to the S shuttle train (color grey) to 42nd Street-Grand Central.Trains on the New York City subway system operate continuously. There are schedules you can download, but they're often inaccurate.
No, but it does stop at penn station and you can take the subway
No, the AirTrain takes you to and from the subway, and the subway takes you to and from Penn Station. Take the AirTrain to the Howard Beach Station, and get the Manhattan-bound A train (of the A-C-E, the blue line) to 34th Street-Penn Station.
No, they are two separate train stations, located in two different places. New York City's Penn Station (there are other Penn Stations in the U.S.) is located at West 33rd to 34th Street, between 7th and 8th Avenues. Grand Central Station and Terminal is located at East 42nd Street between Park and Lexington Avenues. Grand Central Station is the subway station, where you can get the 4-5-6 (the dark green line), the 7 (the purple line) or the S shuttle train to Times Square. Grand Central Terminal is the railroad terminal for non-subway trains running in and out of New York City.
Take the Downtown 2 or 3 trains (of the 1-2-3, the red line) from 34th Street-Penn Station to Wall Street (at William Street).
MetroNorth from Poughkeepsie goes to Grand Central. Amtrak from Poughkeepsie goes to Penn STation. It's very easy to take MetroNOrth to Grand Central and then get the subway to Penn Station.
They are both on the New York Subway system, but you'll have to transfer. The best way to get from Grand Central Station to Penn Station is to take the "S" train to Times Square and then take the "1", "2", or "3" trains heading Downtown. The 34th Street stop is by Penn Station. The subway map calls the A, C, E 34th Street stop the Penn Station stop, but the 1, 2, 3 34th Street stop is just as close; walking from the "S" train to the A, C, or E would take awhile since there is a lengthy walk from the S stop on 7th Avenue to the A, C, E stop on 8th Avenue. It's not worth that.
Yes, the A express train goes from Penn Station to Chambers Street. It's 4 stops (unless you are traveling at night, when the A runs local). The Chambers Street station on the A and C trains is at Church Street.
Take the Uptown E train (of the A-C-E, the blue line) from 34th Street-Penn Station to Court Square.
You don't need to take the subway to get from Penn Station to 28th Street and 8th Avenue (which is in the Chelsea neighborhood, by the way, not in Greenwich Village or Greenwich Connecticut). Penn Station is at 34th Street between 8th and 7th Avenues. So you can just walk there.
Take the Downtown A-C-E (the blue line) from 34th Street-Penn Station to West 4th Street.Transfer at West 4th to the Downtown B or D trains (of the B-D-F-M, the orange line) to Grand Street (at Chrystie Street).Then walk 4 blocks west on Grand Street to Mulberry Street: Chrystie Street ---> Bowery ---> Elizabeth Street ---> Mott Street ---> Mulberry Street.
Take the Uptown 1-2-3 (the red line) from 34th Street-Penn Station to 42nd Street-Times Square, and transfer there to the N.