Sonic Advance 3 features an automatic saving system, ensuring that your progress is regularly recorded. You don't need to manually save your game, which allows you to focus entirely on the action. The game saves after completing each Act within a Zone, after defeating a boss, and after collecting a Chaos Emerald. This means you can safely turn off your Game Boy Advance (or emulator) after reaching these checkpoints without fear of losing significant progress.
While the auto-save is convenient, it's important to remember that if you quit mid-Act, your progress within that specific Act will be lost. You will restart at the beginning of the Act when you next load your game. Therefore, it's always a good idea to aim to complete an Act before taking a break. The game also saves your collected Chaos Emeralds and any unlocked characters, so once you've obtained them, they are permanently recorded.
The game also keeps track of your high scores and 'A' ranks for each Act and boss battle. These records are also saved automatically, encouraging replayability and mastery of each stage. So, feel free to experiment with different character combinations and strategies, knowing that your achievements are being preserved.