Most Greek children never went to school at all. Girls, to begin with, always stayed with their mothers until they were married, either at home or working in the fields. Slaves, whether boys or girls, also could not go to school, and many children in ancient Athens and Corinth and other Greek cities were slaves. Any boy who was poor, even if he was free, also could not go to school: his family could not afford to pay the teacher, and besides they needed the boy's work at home. There were no public schools.
Still, people who could spare the money did try to send their boys to school, because without learning to read and write and generally becoming educated, boys could not hope to participate in politics when they grew up.
Greek schools were small. They had only one teacher and about ten or twenty boys. Boys began going to school when they were about 7 years old, and went until they were about 13.
In school, boys learned to read and write, and also memorized large amounts of Homer's Iliad and Odyssey. They learned to play the lyre (the kithara) and the pipes (the aulos), and to sing.
To find out more about Greek schools, check out these books from Amazon.com or from your library:
Eyewitness: Ancient Greece , by Anne Pearson. For kids.
Ancient Greek Children (People in the Past Series-Greece), by Richard Tames (2002). For kids.
Children and Childhood in Classical Athens, by Mark Golden (1993).
Ancient Literacy, by William V. Harris (1991). Groundbreaking. Argues that there was much less literacy in the ancient world than people have assumed.
A History of Education in Antiquity, by Henri Marrou (1948). Nothing better has come along yet, so this is the old stand-by.
The Gymnasium of Virtue: Education & Culture in Ancient Sparta, by Nigel M. Kennell (1995).