Boys of wealthy families attended schools called "palaestra" where physical education was the focus. They likely would have worn tunics or loincloths for these athletic pursuits.
Girls received informal home education from mothers or tutors on domestic matters such as weaving, reading, singing, gymnastics, dancing, playing the lyre, managing servants, handling finances, and supervising slaves. The clothing they would have worn during these lessons would have been ordinary daywear like tunics.