When you first load the game, you will jump straight into the board game portion.
When you load the game a second time, you will jump into a board game again. Completing the game will unlock the Balloon Island mini-game and the plaza where you access all other parts of the game.
When you play the board game a third time, you will be then given the option to use Amiibo Cards to add villagers into the board game. You will also be given the option to use tickets to purchase public works (such as Brewster's coffee shop) to add to the board. Adding works can open up other paths to travel.