Yakuza 3 features a manual save system. You cannot save anywhere at any time. To save your progress, you must locate a phone booth. These are scattered throughout Kamurocho and Okinawa. Look for the distinctive green phone booth icon on your mini-map.
Interacting with a phone booth will bring up the save menu. It's highly recommended to save frequently, especially before major story missions or challenging boss fights. The game does not have an autosave feature for general progress, though some story segments might create temporary checkpoints. Losing progress due to not saving can be frustrating, so make it a habit to pop into a phone booth whenever you pass one or complete a significant objective. There are also phone booths located within certain establishments, such as the Serena bar.