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 intersection of Canal and Varick Streets. Walk three blocks north on Varick to Broome Street (Canal Street ---> Grand Street ---> Watts Street ---> Broome Street).
To get from Penn Station to Worth Street, you can take the 1 subway line. Board the 1 train at 34th Street-Penn Station and travel downtown. Get off at the Chambers Street station, which is close to Worth Street. Alternatively, you could take the A or C train from 34th Street-Penn Station to Chambers Street for a similar route.
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.
To travel from LaGuardia Airport to Penn Station using the subway, you can take the M60 bus from the airport to the Astoria Blvd subway station. From there, you can transfer to the N or W train and take it to 34th Street-Herald Square station, which is near Penn Station.
To get to the Hayden Planetarium from Penn Station, take the A, C, or E subway line from 34th Street-Penn Station. Transfer at 42nd Street-Port Authority Bus Terminal to the 1 subway line, then travel uptown to 79th Street. From there, it's a short walk to the American Museum of Natural History, where the Hayden Planetarium is located.
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).
To get from Penn Station to 54th Street in New York City, you can take the 1, 2, or 3 subway lines from 34th Street-Penn Station. Transfer at Times Square-42nd Street to the N, Q, R, or W trains heading downtown. Alternatively, you can also walk a few blocks to catch the E train at 34th Street-Penn Station, which will take you directly to 54th Street.
To travel from Penn Station to 851 Grand Concourse in the Bronx, take the 2 subway line from 34th Street-Penn Station station towards Wakefield-241st Street. Transfer at the 149th Street-Grand Concourse station to the 4 or 5 line, and then continue to the 167th Street station. From there, it's a short walk to your destination at 851 Grand Concourse.
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 Uptown 1-2-3 (the red line) from 34th Street-Penn Station to 42nd Street-Times Square, and transfer there to the N.
Penn Station is at 34th Street so it is 20 blocks away from 54th St. You would take the subway two stops north.