Speaking to the Greeks-

Ancient Greece was a collective of city states, the most known likely being Athens, Sparta and Thebes. Each society functioned according to different values and societal goals.

In the 4th and 5th century BCE, Children in Sparta were considered to be property of the city state, and boys went into intense training at the age of 8 where only 1/4 would make it to the age required of a fully developed soldier. Girls in Sparta often exercised with boys, and were more "liberal" in their expectations, training naked among boys and having more of a voice in society. They were expected to be in peak physical condition so their children would be strong as well.

In the same time period, Athen's focus was more on education and political life. It was considered the obligation of every citizen to be politically active. Children were tutored intensely and a separation between boys and girls was clear. Nudity between genders was unheard of and the "best" wife was one who was seen and not heard.

As for Roman children, I am much less knowledgable, and will let someone else step in.