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West Street subway station was created in 1896.
Penn Station is at 34th Street so it is 20 blocks away from 54th St. You would take the subway two stops north.
Walk west to Lexington Avenue and north to 53rd Street, and take the Downtown E train (of the A-C-E, the blue line) from Lexington Avenue-53rd Street to 34th Street-Penn Station.
Probably the best way to get there is to take the C or E train from Penn Station to the 23rd street stop or the 1 train to the 28th street stop. However, to be completely honest, you could walk from Penn Station to 208 W 23rd Street in around 10-15 minutes.
Take the Uptown E train (of the A-C-E, the blue line) from 34th Street-Penn Station to 53rd Street-5th Avenue. Fifth Avenue is one block west of Madison Avenue.
Take the Downtown A-C-E (the blue line) from 34th Street-Penn Station to West 4th Street. Transfer at West 4th Street to the Downtown F train to Delancey Street-Essex Street. Delancey Street is one block south of Rivington Street.
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 A-C-E (the blue line) from 34th Street-Penn Station to West 4th Street (doesn't matter which you take, the A, C or E). Transfer at West 4th Street to the Downtown F train (of the B-D-F-M, the orange line) to East Broadway.You'll exit the subway at or near the corner of East Broadway and Rutgers Street. Walk one block south on Rutgers Street to Henry Street.
Hip, trendy bars in the area around Penn Station are few and far between. You would be much better off taking the subway down to the West Village. The West Village is one of the trendiest neighborhoods in the city, and both the A-C-E subway line (the blue line) and the 1 or 2 trains (of the 1-2-3, the red subway line) run directly between Penn Station and the West Village. If you take the A-C-E, then you will be going from Penn Station (34th Street) to West 4th Street. That is TWO STOPS on the A train, and THREE STOPS on the C or E. If you take the 1 train, then you will be going from Penn Station (34th Street) to Christopher Street/Sheridan Square. That is FIVE STOPS. Do NOT take the 2 or the 3 trains, as they are express trains and do not stop at Christopher Street/Sheridan Square. The West Village is absolutely loaded with trendy bars and restaurants. Just walk around a bit and pick one you like.
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.
Assuming you meant "6th Avenue and 45th Street": the easiest way would be to take the Uptown 1, 2, or 3 trains (the red line) ONE STOP from Penn Station/West 34th Street to Times Square/West 42nd Street. You will exit the subway at around 7th Avenue and 42nd Street. Walk north 3 blocks to West 45th Street. Then turn right, and walk one block east to 6th Avenue.
Take the Downtown 2 or 3 trains (of the 1-2-3, the red line) from 34th Street-Penn Station to Chambers Street (at West Broadway).You could also take the 1, but you don't want to, because the 1 is a local train and there are many stops from Penn Station to Chambers Street. It's only 2 stops on the 2 and 3 trains.