Hi Maximilian.
The game should save your progress into your playthrough each time you make a decision and when you close the game. While the game is intended to be played in real time, a Skip Mode toggle exists in the settings menu to disable this. (available once reaching the normal or true ending on mobile, or from the beginning of the game on PC) Skip Mode removes the waiting times between messages (excluding typing) allowing you to complete a single playthrough of the game in a single sitting. If you have any further questions, be sure to let us know! (And sorry for taking so long to get back to you, been a busy few months on our end)