You save the game by talking to this dog, Save-berus. You can find the dog in several locations around Dream Island. There are no save points in the tutorial/start of the game.
If you want to load your game without going to the title screen, press (Y) on the controller or [s] on the keyboard and use the Phone to call the dog (saveberus).
I know this is a very retro setup, but it's part of the feel I was going for.