The Gate is the portal to access most labyrinths. At the beginning of the game you can select the labyrinths, but once you are about to meet the Forest Hermit, you will no longer be able to select the location and will instead be randomly transported to any of the labyrinths.
At that point, the stables will be available as an alternative, with the difference that you will have to pay a fee to select the stage.
Besides this, the Colosseum will be available later. Here you'll be available to fight against other players in a versus mode.