In the main menu (After the splash screen), I only see my save game, Mystery Gift, and Live Competition. How do I start a new game?
You first need to delete your existing saved game.
To delete a save game, hold
UP + B + X in the title screen (the screen with Solgaleo or Lunala). This will also give you the option to re-pick the language setting of the game after you delete your save.
After you delete your save, you will be able to start a new game.