The Adventurer's Guild is a small house next to the entrance to the Mines in Stardew Valley. This guild also houses a shop where you can buy weapons, rings, and boots, so it's smart to unlock it if you want to progress down into the Mines.
The first step in unlocking the Adventurer's Guild is to reach level 5 in the Mines. Now check your mailbox the next morning. This will give you the quest "Initiation". Do the quest by slaying 10 slimes of any color to complete it. This should be pretty easy to do.
Once you have completed the quest, Adventurer's Guild should be open for you. Just head there between 2PM and 10PM to check out the Shop and the Monster Eradication Goals.