It is not possible to "save" during the middle of a session and continue later on. This is because of the games Arcade style of play - you play through all 10 levels trying to rank in the high score...
You get medals by getting a certain amount of hits in each level. The amounts are: Corneria: 150 Meteo: 200 Fortuna: 50 Sector x: 150 Titainia: 150 Blouse: 150 Venom: 200 Katina: 150 Solar: 100...