Take the 1 train (of the 1-2-3, the red line), or the 2 late nights, to Houston Street. Even though it's called the Houston Street Station, you will get off on Varick Street, between King and Houston Streets.
If you're going further south on Varick Street, then take the 1 (or 2 late nights) to Canal Street (at Varick Street).
You could also take the C train (of the A-C-E, the blue line), or the A late nights, to Spring Street. It will let you off at the intersection of 6th Avenue and Spring Street. Sixth Avenue runs north, and Spring Street runs east. You need to walk one block west on Spring Street to get to Varick.
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).
Take the Downtown 1 train (of the 1-2-3, the red line) from 34th Street-Penn Station to Canal Street (at Varick Street). Then walk 1 very short block north to Grand Street. The address 75 Varick Street is approximately where Grand Street is.
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.
There is no "121 Houston Street." There is a 121 East Houston Street and a 121 West Houston Street.The closest subway station to 121 East Houston Street (at 2nd Avenue/Chrystie Street) is the F train (of the B-D-F-M, the orange line) to 2nd Avenue-Houston Street.The closest subway stations to 121 West Houston Street are the C or E trains (of the A-C-E, the blue line) to Spring Street (at 6th Avenue) or the 1 train (of the 1-2-3, the red line) to Houston Street (at Varick Street).Coming from either station, you'd walk up to Houston Street and then east to 121 West Houston Street. Spring Street is a couple blocks south of Houston, and the Houston Street Station on the 1 train is actually on Varick Street between King and Houston Streets.
Take the Downtown 1 train (of the 1-2-3, the red line) or the 2 late nights, from 34th Street-Penn Station to Houston Street (on Varick Street between Houston and King Streets).Walk 0.5 blocks north to Houston Street, and 0.5 blocks east to 209 West Houston Street (between Varick Street and 6th Avenue aka Avenue of the Americas).
You can take the Downtown 1 train (of the 1-2-3,the red line) from 42nd Street-Times Square (at 7th Avenue) to Canal Street (at Varick Street). This station is on the West Side of Manhattan.Or, you can take the Downtown N-R-Q-W (the yellow line) from 42nd Street-Times Square to Canal Street (at Broadway). This stop is further east than the Canal Street Station on the 1train. It will drop you off pretty much exactly in the center of Manhattan.
If you go down the street there should be one.
A useful answer to this question cannot be given unless you specify where in Brooklyn you are coming from. Brooklyn is slightly less than 100 square miles in area, and 8 of New York City's 9 subway lines (not including the shuttles) run through it. The directions will be completely different, depending on which subway line is closest to your starting point.
The closest subway stops are the 18th Street stop on the 1 train (of the 1-2-3, the red line), at 7th Avenue, and the 23rd Street stop on the C and E trains (of the A-C-E, the blue line), at 8th Avenue.
Market Street Subway was created in 1978.
Tremont Street Subway was created in 1897.
West Street subway station was created in 1896.