By their parents at home.
They went to a tutor who taught them how to play an instrument. They also learnt to sing from their parents.
In ancient Greece music was very important. Music was a part of education in ancient Greece, boys were taught music at age six. In ancient Greece, mixed-gender choruses performed for entertainment, celebration and spiritual reasons. Instruments included the double-reed aulos and the plucked string instrument, the lyre, especially the special kind called a kithara.
The most important contributions of ancient Greece to medieval music are probably the modes or ancient scales used in both. There is a link to an article on them below.
music writing and art
Calliope - my name yay
For escape of mind, stress realief and sernizing
orpheus
Sophists were professional teachers of rhetoric and other subjects , noted for their ingenuity and speciousness
Ancient Greece was known for its art, music, and architecture. They were also known for poetry and writing. All of these things are things that people in western culture emulate.
Girls in ancient Greece generally received no formal education; it was a culture that believed education for boys and young men was much more important. However, there were exceptions. Certain wealthy upper-class families sometimes tutored their daughters at home; and although this training was mainly intended to develop a girl's skill in the domestic arts, there is evidence of some lessons on literature and music. But this was not the norm, and most girls did not get this kind of opportunity.
I think the general consensus is that our concepts of scales and notes originated in ancient Greece.
The word music comes from the Greek word (mousike), which means "(art) of the Muses". In Ancient Greece the Muses included the goddesses of music, poetry, art, and dance.