It seems like everyone in The Taming of the Shrew is trying to get some action, which is why a character's relationship to "sex and love" can tell us so much about him or her. Petruchio, of course, is on the prowl for a rich wife (even if she's old and unattractive). His attitude toward marriage is ruthless and his honesty about his desires is almost refreshing in a play where most of the men are after a rich wife.
Kate, on the other hand, says she's not interested in marriage but she gets angry when Bianca refuses to give her the dish on her boyfriends. This reveals a whole lot about Kate. It tells us that she is in fact interested in love but that she objects to the way marriages are arranged and negotiated between men.
Christopher Sly is also an interesting case. When he's duped into thinking he's married to a trophy wife (who is really a young boy dressed up like a noblewoman), his sexual attraction to a cross-dressed boy reveals his ignorance and alerts us to the fact that he's someone to be mocked.
Sly's social status as a "beggar" and a "drunk" makes him the object of disdain when the Lord finds him passed out in front of a tavern. As such, Sly is completely powerless when the Lord decides to teach him a lesson. Like Katherine, Sly is punished for his unruly behavior and held up for public ridicule. Even though Kate is from an upper-middle class family, she's still a woman, and that means she has little power – her transformation from a "shrew" to good wife is inevitable, even though it may not be genuine.
Perhaps more than anything else, a character's relationship to speech defines who he or she is. Kate is labeled a "shrew" for her acid tongue, whereas Bianca is seen as an attractive, ideal woman when she demonstrates obedience and silence (this turns out to be a mere façade). Sly's language is also telling when he misquotes lines from famous plays and doesn't know the proper term to use when addressing his "wife." It seems like a character's language reveals how much or little power that figure has.