In Baptista's house, Katherine is teasing Bianca. She has Bianca's hands tied and asks her which of her suitors she likes the best. Bianca says she is not in love with any of them. Katherine says perhaps Bianca desires a wealthy husband, in which case Gremio would do just fine. Katherine hits Bianca.
In one of the few scenes where Katherine and Bianca interact, Katherine takes out her anger against traditional female roles and an economic understanding of marriage on Bianca, who—as an obedient woman and valuable bride—exemplifies both.
Baptista enters and is upset to see Katherine abusing Bianca. He unties Bianca's hands and sends her off to sew. He chastises Katherine, saying Bianca has done her no wrong and has said nothing against her. Katherine replies, "Her silence flouts me, and I'll be revenged!" (ii.1.32) She accuses Baptista of favoring Bianca over her and leaves angrily.
Gremio enters with Lucentio (disguised as Cambio). Petruchio enters with Hortensio (disguised as a tutor named Litio). Tranio (disguised as Lucentio) enters with Biondello. Petruchio introduces himself and tells Baptista he is interested in Katherine. He introduces "Litio" (really Hortensio) as a teacher of math and music, who can instruct Bianca. Baptista thanks him for finding the teacher, but is skeptical that Petruchio really wants Katherine.
It is almost hard to keep everyone's identity and disguise straight, showing how the play's proliferation of disguises and performing raises questions about one's "real" identity.
Gremio interrupts to introduce "Cambio" (really Lucentio) as a teacher of Greek, Latin, and other languages. Baptista thanks him for the teacher, and then asks who Tranio is. Tranio introduces himself as Lucentio, and says that he is a suitor for Bianca. He presents Baptista with some Greek and Latin books as a gift. Baptista is pleased and says that he knows Lucentio's father. He has a servant lead the two "teachers" inside to Bianca.
Baptista thinks that Bianca is receiving an education in languages and music, safe at home, but she will really receive a practical education in romantic courtship.
Petruchio discusses the dowry for Katherine and assures Baptista that he is strong enough to make Katherine yield to him. He claims, "I am rough and woo not like a babe," (ii.1.144).
Hortensio enters (still disguised as the music-teacher Litio), pale and with an injury on his head. He reveals that he tried to teach Katherine how to play the lute, but she got frustrated and broke the lute over his head. Petruchio is amused and says he loves her even more than before. Baptista tells "Litio" (Hortensio) to try teaching Bianca instead. He goes inside to send Katherine out to meet Petruchio.
Hortensio's comedic injury shows that Katherine is not interested in learning traditionally proper female activities like music, and is not afraid of using physical violence. Petruchio seems to like Katherine for this, whether because he enjoys a challenge in wooing/taming her or because he actually wants a marriage of two like-minded (that is, stubborn) people.
Alone on-stage, waiting for Katherine, Petruchio plans how he will woo her. Regardless of what she says, he plans to compliment her and act as if she is being polite and kind. Katherine arrives and Petruchio praises her as beautiful and fair. Katherine tells him to leave, and the pair engage in a long exchange of witty word-play, each using punning to twist the other's words.
Katherine strikes Petruchio and he threatens to "cuff" her if she does so again. The two continue to spar with words and wit. Petruchio says that he has heard that Katherine is "rough, and coy, and sullen," (ii.1.257) but that she is actually pleasant and sweet. Katherine is frustrated by his absurd praise. Petruchio tells her that Baptista has agreed to make him Katherine's husband. He tells her, "I am he am born to tame you, Kate, / And bring you from a wild Kate to a Kate/ Conformable as other household Kates," (ii.1.291-293).
Petruchio describing Katherine as pleasant and sweet (exactly the kind of personality she refuses to have) irritates her even more than the usual insults of characters like Gremio. His use of the imagery of taming (punning on the similarity of "Kate" and "cat") again shows his misogynistic desire for complete control over his wife.
Baptista enters with Gremio and Tranio (disguised as Lucentio). Katherine complains to her father that he has wed her to a lunatic. Petruchio says that Katherine is modest, patient, and chaste, and announces that they have agreed to marry on the upcoming Sunday. Katherine quips that she'll have him hanged on Sunday. Petruchio says that he and Katherine have agreed that she will pretend to still be ill-mannered and upset in public, but that in private she and he are getting along nicely. Baptista approves of the wedding. Petruchio and Katherine leave, but through different doors.
Petruchio's claim that he and Katherine are pretending to hate each other seems to be an obvious lie—but with all the pretending going on in the play, how certain can we be? In any case, Baptista is easily persuaded, as he simply wishes to marry Katherine off to someone, regardless of how she feels about the union. Katherine leaves through a different door than Petruchio as a minor act of defiance against his control.
Baptista now sets his mind to figuring out who Bianca's husband will be. Gremio claims that he loved Bianca first, but Tranio says he loves her more. Baptista breaks up their dispute by saying that whoever offers the better financial situation for Bianca will get her. Gremio lists all of his property and riches. Tranio (speaking as Lucentio) catalogues all the wealth of Lucentio's father (which is even more than what Gremio can offer), to whom he is the sole heir.
Baptista decides who will be Bianca's husband based solely on the suitors' financial situations. Since Lucentio's wealth is dependent on inheriting his father's property, the ideas of marriage as a union of families and as an economic exchange are shown to be closely related.
Baptista says that he will give Bianca to Lucentio on the Sunday after Katherine and Petruchio's wedding, provided that Lucentio's father guarantees Lucentio's inheritance. Otherwise, Bianca will be married to Gremio. Baptista leaves. Gremio is confident that Lucentio's father will not turn over all his property to Lucentio, and he leaves. Left alone, Tranio plots to find someone he can disguise as Lucentio's father.
Again, Baptista's approval of the marriage between Lucentio and Bianca is contingent upon a financial guarantee from Lucentio's father. Baptista does not stop to think what Bianca might think about this union. Meanwhile, Tranio continues his clever plotting, driving the plot of the play forward.