The lord is introduced praising his (well-trained) hunting dogs. In retrospect, of course a person like that would enjoy having plays in which women are tamed performed.
When Katharina hits Petruchio in the fiery and hilarious dialogue of their first meeting, he warns her that if she does that again, he'll retaliate. Kate answers with a remark about how if he beats her he will be no gentleman. Now that was really bold for Kate: beating Hortensio, who after all is (or pretends to be) a servant while she's the lady of the house is a thing of little consequence, but striking the suitor who's equal to her in status and approved by her father in a very patriarchal society is quite different. Petruchio would be in his right hitting her back and her dad would only say "serves you well". Considering how Petruchio, despite his bombastic language and threats, never resources to physical aggression, Kate's instinct seems right from start: he's not the kind who would hit a girl. It adds to the brilliance that she hits him explicitly saying that she will try if he's a gentleman as he claims: he passes the test.
Katharina, despite all her protests, is attracted to Petruchio at first sight. This is clear from the marriage scene: though she claims to be marrying against her will, when Petruchio is late and she fears that he might be a crazy jerk who proposes to women everywhere just for the lulz, she's genuinely panicking and in anguish. It's not just the humiliation of being left at the altar, she's heartbroken and crying. "Would Katharina had never seen him though!" she bursts using the third person in truly Shakespearean manner, which, in this case, denotes a highly emotional state.
Shakespeare's The Taming of the Shrew is arguably funny and light-hearted when you first look at it, but the protagonist seems to be developing Stockholm Syndrome by the end...
There's a lot of debate over if this play is indeed an affirmation of sexist ideas or a subversion of them, or something in-between. The critical point seems to be if Katharina's famous last speech is intended to be serious or if it's an ironic internal joke between her and Petruchio. The fact that Petruchio and Katharina are having the time of their lives in the last scene, shocking everybody with her transformation and the "foolish duty" demonstration, and Katharina's discourse on the feminine submission is so exaggerated and excessive, seems to support the latter interpretation.
Willy's good old Ode to Wife Beating? I think that goes in several places, but it's very apparent even slightly into the play.
Or Ode Against Wife Beating, given how Petruchio emphasizes how he can tame his wife without beating her. Still not compatible with modern values, but...
Remember also that the Marital Rape License would have been in effect when this was written...
Though the concept of marital rape was unheard of in the 16th century, it's evident that Petruchio does not want to have sex with Kate before she warms up to him and consents. Everybody remembers the charming "kiss me Kate" scene, when Petruchio tries to manipulate Kate into giving him a kiss, but in an inversion, it's she who learns how to manipulate her childish and hyperbolic husband – both see through the other's actions and delight in them. It foreshadows the full sexual consent later. Almost the last words he says in the play are "Come, Kate, we'll to bed", i. e., they will have their nuptials at last!
