The account usually gets automatically connected to Google Play/Game Center/Huawei AppGallery/Samsung Galaxy Store when you first enter the game, if you gave your consent to this. You can check if your account is connected in the game settings. If you see the button “Disconnect”:

it means that your game account is connected and your progress is automatically saved.
And if your settings window looks like this:

then we advise you to tap the button “Connect” (the left one of the two buttons). 
Please note that if the button does not change from "Connect" to "Disconnect" when clicked, and if the game suggests loading some other progress instead (which means - not saving the current progress), then the actions in this case are described in a separate article