In its early days, when Rome was a kingdom, kids did not go to school. Education took place in the home and was done by the family. If a family had someone who knew how to read and write, the boys were taught how. They were also taught how to be warriors. Finally, they were taught how to manage the farm or business and how to behave in society. All this teaching was done by other males in the household.
Girls were taught by the females in the household. They were taught how to run a household and how to be a good wife.
If they could afford it the family might hire a tutor to teach math and oration, but mostly the teaching was by the family.
This changed during the republic. The Romans saw how the Greeks taught their children using paid teachers to educate groups of students. The Romans figured that this was a pretty good system so they adopted it. However, school was not free. You had to pay the teacher so poor children did not go to school.
Teachers taught more than just reading and writing. They also taught math and Greek literature. But the main subject was Oration or public speaking.
School started before sunrise with students working using candles or oil lamps. They took a break for lunch and siesta, then worked again until late afternoon.