To save your progress, you'll have to install the iTunes Game Center app on your device from the iTunes store.

To save your account:
- Enter the game settings by tapping/clicking on the cogwheel placed in the upper right corner of the main game screen.
- Press the "connect" button.

Once you've done it, the progress on your account will be saved on the iTunes Game Center.
If the game is reinstalled or played on another device, you'll have to redo all the steps to load your saved profile up. The device must also be synchronized with the same Apple ID profile used to save your main profile.
After choosing the "connect" option, a window will appear, letting you choose your main profile. There, you will have to choose the account which will be loaded on the device. The other profiles will be lost.