Many people today think of Athens as an enlightened society and it may have been that it some ways. Nevertheless, it is in Sparta, not Athens, that women were taught to read and write. Like their male counterparts, however, much of their education focused on physical conditioning. Just like men, women were taught dance and gymnastics and competed in track and field events and staged battles. Also like men, the competed in the nude, even in front of their male counterparts.
Girls in Sparta did not receive the kind of domestic training (spinning & weaving, etc.) as women in Athens because domestic work was reserved for the helots, slaves, in Sparta. They did need to spend much time raising their children either since professional nurses usually cared for younger children and boys entered the Agoge at the early age of seven years old. While there was no female equivalent to the Agoge, girls were subject to tests of physical strength just like the men were.
Perhaps the one area in which Spartan culture was particularly paternalistic was in marriage. Women had literally no say over who they married and may have never met their husband prior to their marriage. Nor was there much ceremony attached to marriage in Ancient Sparta. When a man wanted a wife, he would abduct her in the night and forced her to be his wife. In fact, anyone man could abduct any woman to be his wife, even if that woman was married. No doubt some husbands were willing to fight over their wives, but polyandry was not uncommon in Ancient Sparta.
In other ways, however, women had more rights over their marriage than most women in the Ancient world. For example, they could buy and own property on their own. They could also marry another man if their husband died or was simply away from home for too long. At the same time, they were expected to maintain their husbands property. This meant that they had command of the helots and that they had to be ready to defend their property if needed. That is why the Spartans insisted on such extensive physical training for their girls. Those Spartan women took their training so seriously that they are often depicted as warriors in ancient illustrations.