How to let the player save progress

This is known as serialization. There is essentially two parts to this question, how to serialize data, how to do file I/O. When I did serialization for my game I used PlayerPrefs which worked quite well. If you specifically want to write out to file with your own file format specification that is another matter which I don't have any experience with in Unity.

With serialization the first step is to determine what part of the game state is important to save and load, and which state can be easily thrown away.

For example: the player's health, the player's money, the player's position are things you probably want to save. The state of a particle emitter, or the soundsystem are things which you can happily discard.

Once you have determine what to save, now you just have to save them. A common approach is to have every object know how to save itself. So every object which contains state you want to save would have a function called something like 'writeSaveFile' and a matching read, 'readSaveFile'. Inside those functions you perform the serialization, making sure that the save and load function match exactly.