Shakespeare's The Taming of the Shrew, a play within a play, is about a man who wants to control his woman, when his woman has completely different ideas about the arrangement.
Shakespeare's version was "revised" in Cole Porter's 1948 Kiss Me, Kate. (It was the first musical to ever win Best Musical at the Tony Awards, in 1949). It was made into a film of the same name in 1953.
In 1999, it was the plot to the movie Ten Things I Hate About You, and the structure of the original has been used also in television shows. For example, in 1986, a Moonlighting episode ("Atomic Shakespeare") starring Bruce Willis and Cybil Shepherd satirized the play with a few new twists (including a rock 'n' roll number: "Good Lovin'").
In all of these versions of Shakespeare's play, the reader sees the age-old battle of the sexes that drives the plot along. Petruchio wants to marry Katarina, however, she will have no man. Her father grows weary of her shrewishness, and Bianca, Katarina's younger sister may not wed until her older sister is married. So Hortensio arranges for Petruchio to wed Katarina.
By the end of the play, Katarina has come around to Petruchio's way of thinking, it would seem. However, it is possible that she merely understands that everyone answers to someone: all people defer to royalty (and even he/she answers to God).
However, Katarina's speech is intelligent and wise, while the women —and men—around her sound foolish, even Petruchio. When she orders the other wives to be submissive to their husbands (following the directions of her husband), she shows that she is in some ways far superior to the men by which she is surrounded, and though she may seem subdued, she really is the better person.
In terms of what we learn, it may be that there is a time and place for everything: knowing when to speak up and knowing when to be silent. We also may learn that just because one person is more powerful than another, it does not necessarily prove him or her of superior intellect or grace. This is what we witness of Katarina in this play.