There is an autosave feature on Subnautica mobile : the game saves when you're opening the PDA. You can trigger an autosave every 2min30 !

To save your game manualy while playing, click on the icon with three bars at the top right of the screen, then click "Save". You can also click "Quit": the game will ask you to save your game before returning to the main menu. 

If you switch between apps, the game will pause and you can resume your game.

If you exit the game without saving and close the app completely, you will be prompted to resume or discard the auto-save the next time you open your save.