Vault 111 is the location where you begin your game and you'll have a chance to explore it during your first visit. You should still consider returning to this location later in the game and one of the motivations for that is a chance of getting a unique weapon called the Cryolator which has been described on the next page of this chapter. You can reach Vault 111 by travelling on foot or by using fast travel. Once you get there look around for a small booth seen on the screen above. Press the red button to activate the elevator.
You don't have to wait for the elevator to show up. As soon as you've pressed the button stand on the large round platform (the same one as in the prologue). The elevator should start moving automatically and it'll take you to the vault.