Wollstonecraft doesn't waste a whole lot of time in getting to the point in A Vindication of the Rights of Woman . She says from the get-go that humanity's greatest gift is its ability to reason. And since men and women are born with the same ability to reason, women should enjoy just as much education, power, and influence in society as men do. The only reason women don't seem as smart as men, she says, is because they aren't given the same education. The one thing she's willing to admit is that men might have an advantage in physical strength. But in a modern civilization, this advantage shouldn't really mean anything. For a gentleman living in Wollstonecraft's time, there were very few (if any) occasions in life where he would be called upon to use all of his strength.
Once she gets into her argument, Wollstonecraft goes after some writers who have claimed that women's education should focus solely on making young women pleasing to men. In other words, popular opinion in Wollstonecraft's time states that women shouldn't busy themselves with too much reading or studying. They should focus on dressing nicely and being quiet.
Wollstonecraft tears these arguments to shreds, saying that they end up causing a lot of social problems. For example, how can people expect a woman to raise children well if she has no education and no ability to reason? Further, how can women be moral and virtuous if all they're ever taught is how to look moral and virtuous? This kind of education focuses only on appearances and makes women totally superficial.
As the book continues, Wollstonecraft argues that education should be available equally to both boys and girls regardless of how wealthy their families are. That's why she thinks that there should be a national public school system that is free for children up to a certain age. That probably sounds familiar; it's a lot like today's public school system.
Wollstonecraft closes the book with one last flurry, summing up all the arguments she's made and showing once and for all that there's no possible way to support the oppression of women without being a bully and a tyrant. In the end, Wollstonecraft states that a future with educated women will be much brighter than a future without them.