As with other farming simulators like Stardew Valley and Harvest Moon, Harvestella auto-saves your game data when you rest at night. However, there’s also a way to manually save your game, but it’s not available until halfway through the first dungeon.
The option to manually save your game becomes available when you get your hands on Motus Magicite. This is given to you by Dianthus, a character who will eventually join your party. However, Dianthus doesn’t give you Motus Magicite until you make it about halfway through the first dungeon you enter.
It will take you several real-world hours to reach this point in the game. Why Square Enix went this route is unclear, but it certainly makes it difficult to seamlessly start the game with breaks shortly after.
Once you’ve obtained the Motus Magicite, you can interact with save points—the glowing orbs scattered around the world. As an added bonus, these save points also act as fast travel hotspots, which is greatly appreciated given how much there is to do in such little time.