Yes, Penn Station is close to Grand Central Station. They are less than a mile apart.
But, if you are asking because you are planning on walking or taking a bus or taxi from one to the other...don't. You can just take the subway: take the Uptown 1-2-3 (the red line) one stop, from 34th Street-Penn Station to Times Square. Then transfer at Times Square to the S shuttle train (color code grey) east across 42nd Street, from Times Square (West 42nd) to Grand Central (East 42nd).
You can also take the Uptown A-C-E from Penn Station to Times Square, but this will require you to walk through a long, hot, smelly underground tunnel. Since the Times Square Station is at 7th Avenue, while the A-C-E line runs up and down 8th Avenue, the A-C-E does not stop at Times Square. It stops at 42nd Street-Port Authority Bus Terminal. The underground tunnel connects the Times Square Station to the A-C-E line, so you can transfer without exiting the subway system.
So, it is preferable to take the 1-2-3 from Penn Station to Times Square, so you won't have to walk through the tunnel. But if you can't find the 1-2-3 or something (or if the A-C-E just happens to be right there when you get off your train) you CAN take it to Times Square.
Bridge Street subway station was created in 1896.
Buchanan Street subway station was created in 1896.
Merkland Street subway station was created in 1896.
Merkland Street subway station ended in 1977.
West Street subway station was created in 1896.
Take the subway eastbound from the Amtrak station (30th Street Station) to 11th Street.
If you are taking Bloor-Danforth subway line, then nearest subway station is Bay station, which is 1.8km north from 610 Bay Street. On Yonge-Unitversity-Spadina, Dundas station is closest to 610 bay street (300 m east)
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.
Walk two blocks to 13th & Market Streets. Take the subway from 13th Street to 30th Street 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.
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.
You don't have to take the subway; it's 2 blocks. Penn Station is at 34th Street between 8th and 7th Avenues, and 520 8th Avenue is between 36th and 37th Streets.