The Speech Club is a club at McKinley. Like many other clubs, Rachel joined the club to get into as many Thunderclap pictures as possible, so she could "practice for the paparazzi".
Not much is revealed about the Speech Club. When Rachel appears in it in the Thunderclap, there are a total of four members, including her. It is assumed that the club trains its members in speaking publicly, and that they take part in competitions. They may have also won a few as well, judging by the trophy in the club's yearbook photo.
Since the club has not appeared since Season One, it is unknown whether the club is still active. It can be assumed it won't appear again.