The Secret Club's Secret
Examine the back door of the armor shop to start this quest. A password is required to enter, which you likely will not have. (It is unknown if the password can be given without 'learning' it in-game first.) Giving the wrong password means you have to wait a day before you can try again. (It is unknown if this is a real-life day or an in-game day.) You can, however, go 'learn' the password before your first attempt, and safe yourself some time.
To learn the password legitimately, head to the bar at the north end of town (directly behind the Jewelry Shop), and talk to the three women sitting in the back. They will of course notice you and not cough up the password. However, if you'll notice there's a window right above their heads. Head around to the next entrance over to listen in through that window.
You'll learn that the password is GSC♦. (If it doesn't come through, the last character is supposed to be a diamond.)
Assuming enough time has lapsed for you to try the password (or if you read ahead and skipped the first attempt part), go give the password at the door. Head inside to become a member and complete the quest.
This will unlock a much better armor shop for Gerudo Town, which sells the Desert Voe set (which provides heat resistance, with a set bonus of shock resistance if you upgrade all pieces to level 2 or higher), and the Radiant set (which lures stalfos-based monsters to you, and grants a damage boost against them if all pieces are level 2 or higher). Note that the Radiant set requires Luminous stones for purchase in addition to rupees.