In order to prevent some potential problems, the game does not start saving your progress until after you make it to Albert's Shipyard for the first time. If you close the game before that, it will not save any of your data and you will have to repeat the introduction of the game.
Once this point has been reached, the game will begin to autosave when you visit or leave an island and trigger a loading screen. You can trigger manual saves by saving and quitting the game in the pause menu at any point.