Lucentio enters with his servant Tranio. He has just arrived in Padua, eager to study philosophy. Tranio says that he, too, is excited for Lucentio's studies, but he encourages Lucentio to mix his studies with pleasure. He tells Lucentio, "study what you most affect [like]," (i.1.40).
Lucentio arrives in Padua to further his education. Tranio encourages him not to focus exclusively on his studies, unaware of just how far Lucentio will take this advice.
Baptista enters with his two daughters, Katherine and Bianca. Two men, Gremio and Hortensio, enter as well, seeking to woo Bianca. Baptista reminds them that he has decided not to give Bianca away in marriage until Katherine, the ill-mannered older daughter, finds a husband. Gremio says Katherine is "too rough," (i.1.55) for him. Katherine responds harshly to Gremio and Hortensio, and Tranio notes how difficult and badly behaved Katherine seems.
The entrance of Baptista, his daughters, and Bianca's suitors establishes the importance of marriage in the play. Baptista's rule about Kate having to marry before Bianca shows him almost bargaining to marry off his daughters. Katherine is quickly characterized as ill-tempered and not traditionally feminine. But, to some degree, her rudeness is only a response to the harsh treatment she receives from characters like Gremio.
Lucentio, meanwhile, has become obsessed with Bianca's beauty. Baptista tells Bianca to leave and go inside, and she is polite and deferential, in contrast to Katherine's rudeness. Gremio and Hortensio protest Baptista's condition for the marriage of Bianca, but he says that his mind is made up.
Bianca displays the deferential good manners that the male characters expect of subservient women. Gremio and Hortensio compete with each other to marry Bianca, while Lucentio falls madly in love with her.
Baptista says that he will keep only schoolmasters in his house, to instruct Bianca in music and poetry. He asks Gremio and Hortensio if they know of any teachers, then exits, telling Katherine to stay. Katherine is offended at being told what to do, and leaves.
Katherine resists being told what to do by men, rebelling against traditional gender roles.
Gremio insults Katherine, then says that he would gladly find a teacher for Bianca. Hortensio agrees, but also tells Gremio that they should cooperate in finding Katherine a husband, since they both desire it. Gremio jokes that a devil would be a fitting husband for Katherine. Gremio and Hortensio agree to seek a husband for Katherine, so that they may fairly compete for Bianca's hand in marriage. They exit, leaving Tranio and Lucentio alone on-stage.
Gremio again heaps abuse on Katherine, reacting to her refusal to be ordered about by men. He and Hortensio cooperate in trying to fulfill Baptista's conditions for Bianca to be married. In some ways, they seem more interested in the competition with each other for Bianca than in Bianca herself.
Lucentio says that he has suddenly fallen in love with Bianca and is desperate to win her heart. Tranio asks if Lucentio also heard about the arrangement with Katherine, and saw Bianca's rude, boisterous sister, but Lucentio speaks only of Bianca's beauty. Tranio fills Lucentio in on Baptista's condition for Bianca's marriage. Lucentio says he has an idea, and Tranio says that he does, too. Lucentio asks to hear Tranio's idea first. Tranio suggests that Lucentio disguise himself as a teacher and go to teach Bianca in her house. Lucentio acts as if he had the same idea.
Though he lacks a noble education, Tranio appears more clever and observant than Lucentio. Not only does Lucentio not register Baptista's condition for Bianca's marriage, but Tranio is the one who actually comes up with a plan for him to get Bianca.
Tranio coyly asks who will play the part of Lucentio, if Lucentio is the teacher. Lucentio tells Tranio to pretend to be him. The pair exchange clothes, so that Tranio looks like a nobleman. Lucentio's servant Biondello enters and is confused to see Lucentio and Tranio in each other's clothes. Lucentio tells him they have switched identities, because Lucentio has killed a man in Padua and fears retribution. Tranio is thus disguising himself as Lucentio, to protect the real Lucentio. Lucentio and his servants exit.
Tranio cleverly acts as if he doesn't know who will pretend to be Lucentio, letting Lucentio think that he comes up with that part of the plan. Their identity switch offers another example of performance establishing someone's identity, aided by their exchange of clothing. It is also a playful turning upside-down of the social hierarchy.
Meanwhile, Christopher Sly is watching the play (just like Shakespeare's audience). One of his servants nudges him and tells him he is nodding off to sleep. Sly pretends that he is enjoying the play and asks, "Comes there any more of it?" (i.1.261)
As the scene ends, Christopher's reaction reminds us of the fact that we are watching a play within a play. His falling asleep during the play pokes fun at his lower-class background (and is perhaps a slight toward Shakespeare's own audience, which didn't always appreciate his productions).