We often hear about sons in Ancient Greek literature. However, what kind of upbringing could a daughter expect in this period?

I teach a high school myth course and we were just discussing this in class this week. My main source here is Barry B Powell's Classical Myth.

Qualifiers: I'm talking fairly wealthy, Classical Period Athenian families. There is extremely little to draw from on the topic of women as the Classical Athenians seemingly thought very little of and about them. Of women, Thucydides has Pericles say, "Your reputation is great ... when there is the least possible talk about you among men, whether in praise or blame" (2.45.2).

So boys carry on the family name and fight for the glory of the polis then return and run the polis. Girls are bargaining chips for clan alliances through marriage. A girl would spend most of her time in the gynaikeion, the women's quarters in the innermost part of the house. Here she would be taught the skills needed to run a household by her mother, which is economics in its most literal etymological sense (oikos household + nomos law). Overseeing the household includes supervising young males, unmarried females, the slaves and domestic animals, the physical house itself, and its provisions. She would probably also make yarn and subsequently weave it into fabric for personal use and possibly for trade. While a woman might not actually perform any tasks herself besides managing, weaving is a traditional woman's role so she would definitely know how to do it- cf. the most famous of all weavers, that most faithful wife and mother, Penelope. It would be a very rare woman to learn to read and write, though they probably had an oral tradition of storytelling.

Women led a fairly isolated life. They could leave the house, but only with a male relative as chaperone. And when they did go out, it was probably for female specific religious festivals. She probably only remained in her parents' house till about the age of 14, at which time she'd be saved from her own self-destructive nature by marrying. For some reason, there was an impression that a young woman was always a hair's breadth away from flying off the handle and destroying the very fabric of society- cf. the myths of the frighteningly independent and willful young virgin goddess, Artemis. For this reason, she would be taught modesty, obedience, and restraint by a nurse or her mother while at home. In her married life, she would learn how to control herself further through the example of her husband, who would help her fulfill her primary societal duty of bearing children. Only her children could be his legitimate heirs so she's got a fairly important role in the microcosm of the home.

Women also cared for corpses and tended graves as only they could handle the blood pollution imported by touching the body. They purified that same blood pollution after childbirth, too. So while women are not talked of often, they serve as the bookends to every Greek person's life- bringing them into the world and seeing them out.