It's always important to remember to save your game, and saving your game in Stardew Valley is actually pretty different from most other games. There is no quicksave feature or even save points that you need to interact with, and instead you need to head to your bed to save.
Whenever you go to bed in Stardew Valley, it counts as the day being over, and it will first tally all the money you made that day. After this screen tough, it will save the game, and you can tell this by a small green notification text saying so in the bottom of your screen.
There is currently no way to save the game during days, so you need to end the day to save. This can be annoying if you are in the middle of something and need to save and quit quickly, but that's the way it is. So that's how you save in Stardew Valley. Hopefully you won't ever miss any progress in the future now that you know how to save the game.