Act 2 begins with the quarrel of the two sisters but immediately suggests the motive of their quarrel in the favoring of Bianca by Baptista. In her immediate reaction to Hortensio’s instruction Katherine’s independence is established: she won’t be “broken” (like a horse?) to the lute. And this immediately appeals to Petruccio. Their first bout of wit follows after Petruccio’s ringing changes on “Kate.” What follows is a sort of rougher version of Beatrice and Benedick. He seems to surprise, maybe almost overcome her by turning her inside out, insisting on her pleasant and agreeable manner when she is at her sharpest. Even when she strikes him, he doesn’t appear to be angry. Clearly she’s never been treated this way before and it must intrigue her. What he projects on her she seems not to accept, but when he describes their amity from 2.1.295ff, she doesn’t object, even though she’s just called him a “half-lunatic, A madcap ruffian and a swearing Jack.” How does she respond to Baptista’s blessing? Is she simply dumbfounded, or does she accept in some way Petruccio’s entirely fanciful vision of their loving relationship? Much work for an actress here.
Tranio — as Lucentio — and Gremio engage in a bidding war for Bianca, until Gremio confesses himself outbid. Baptista, the entirely conventional comic father will go with the highest bid. And all this ignores the woman’s wishes.
Gremio and Grumio: this could provide some comic business on a modern stage, but did Sh. notice the problem?
Act 2 is a single scene of more than 400 lines. I agree that we get a glimpse of what might be motivating Katherina — as she says to her father, “[Bianca] is your treasure.” But except for that moment, Shakespeare doesn’t seem to be interested in “character” at all. I don’t think Kate is really resentful of or angry at her oh-so-sweet sister, but her exit line — “I’ll be revenged . . . find occasion of revenge” — makes her sound a bit like Malvolio.
The entrance of Petruchio, Tranio, and the disguised “tutors” can only be played broadly, with ridiculous costumes. And it’s really the sideshow. Petruchio’s certainty of his success with Kate sets him up, in our eyes, for a comic fall, but we are pleasantly surprised when he’s more than a match for Kate. Their exchange is a great comic scene, and you’d think it is almost failure-proof. But we have no sense of real people actually exchanging lines or insults. Maybe good actors could convey with their eyes, their tone of voice, their body language, that they don’t mean what they say, and that these two people are actually feeling each other out. Maybe Kate is played by a very good looking actress, and when Petruchio gets his first look at her he realizes that her beauty is a real bonus. (Or would a director have her “dress down” and hide her hair in a scarf?) But we in fact do not yet have any reason at this point to think Petruchio is interested in anything but Kate’s father’s money. Or that Kate has any interest in Petruchio.
The end of the long scene shifts again to the subplot, and Baptista sells his daughter for the “highest dower.” It would be ugly if it were not patently comic, and if the two suitors did not each over-promise.