Please note that once you connect the game to Game Center, you won't be able to disconnect the game from it or use this Game Center account to connect another game to it.
You can connect your game to Game Center to save your progress. There are three possible situations you may find yourself in:
You are connected to Game Center
You may simply proceed into the game - the progress will be saved automatically.
You are not connected to Game Center
You will be asked to enter your login and password upon loading.
The game will be saved automatically upon entering the credentials. You may refuse to do it at the moment and get back to it later - just go to the game Settings.
You are not connected to Game Center and have refused to do so three times
You will need to go the Settings of your device and choose Game Center:
By default, the game will use the currently active Apple ID. In order to use it, simply choose "Sign In".
If you want to use another Apple ID, choose "Use other Apple ID" and enter its credentials. Please note - this operation won't change the current Apple ID of your device. It will only be utilized by the game.
Enter the game and the progress will be saved automatically.
Change the Game Center account
In order to change the account, you need to go to Settings -> Game Center and tap on your Apple ID.
You will be asked to sign out of the current account:
Sign out of the current account and then you will be able to connect to another account.