Petruchio arrives in Padua with his servant Grumio, to see his friend Hortensio. Petruchio tells Grumio, "knock me here" (i.2.8) at Hortensio's gate, meaning knock on the door to announce his arrival. Grumio misunderstands (or pretends to), asking why Petruchio wants him to knock him (that is, hit him). Hortensio enters as Petruchio is angrily hitting his disobedient servant.
Grumio can be seen as either incompetent, failing to understand Petruchio's idiom, or as jokingly clever, pretending not to understand so as to irritate his master.
Hortensio tells Petruchio to end his quarrel with Grumio and asks what has brought Petruchio to Padua. Petruchio says that his father has passed away and he now seeks a wife. Hortensio asks if he would be interested in a wife that is very wealthy, but a shrew. Petruchio says that all he cares about is finding a wealthy wife, and is eager to meet Katherine. He adds that Baptista knew his father and wants to go see Katherine immediately. Grumio is confident that Petruchio can reform Katherine's bad manners.
Petruchio seeks a wife because he feels it's time for him to be married, not out of any kind of love or romantic desire. He is interested in Katherine because of her father's wealth, further showing non-romantic understanding of marriage.
Hortensio tells Petruchio that he must accompany him to Baptista's house, since he is in love with Baptista's younger daughter Bianca, whom Baptista refuses to marry off "Till Katherine the curst have got a husband," (i.2.129). He asks Petruchio to present him (in disguise) as a music teacher, so that he can enter the house and be near Bianca.
Katherine continues to be insulted by virtually all the male characters of the play. Hortensio's planned disguise is another example of all the acts of performance and false identities that pervade the play.
Gremio enters with Lucentio, who is disguised as a schoolmaster named Cambio. Gremio tells Lucentio to teach Bianca only "books of love," (i.2.147). Lucentio promises to plead for Bianca to marry Gremio while he teaches her.
Gremio hopes that Cambio's teaching will make Bianca desire romantic love. Little does he know that Lucentio will try to use his teaching to make her fall in love with him, not Gremio.
Hortensio greets Gremio, who tells him that he is on the way to Baptista's house, to bring Cambio to teach Bianca. Hortensio responds that he has found someone to teach Bianca music. He then introduces Gremio to Petruchio, who he says "Will undertake to woo curst Katherine," (i.2.185) for her dowry. Gremio asks if Petruchio is aware of Katherine's faults and wonders if he can "woo this wildcat," (i.2.198). Petruchio is confident that he can.
Petruchio is again interested in Katherine mostly for her dowry. Gremio's misogynistic comparison of Katherine to a wildcat that must be tamed insults her as animal-like and less than human. The idea of "taming" captures the men's ideal of ownership and control over their wives, and of women in general.
Tranio enters, disguised as Lucentio, with his servant Biondello. Tranio asks the group how to get to Baptista's house. Hortensio asks if he is a suitor of one of Baptista's daughters. Gremio and Hortensio each protest that Bianca is already theirs. Tranio says that his father knows Baptista, and he is not worried that Bianca has other suitors.
Tranio's confident claim that his father knows Baptista reveals an understanding of marriage as primarily about linking together families (and the financial fortunes of those families).
Petruchio informs Tranio that Bianca cannot marry until her older sister, whom he wants as his wife, is married. Tranio agrees with Hortensio that Petruchio's pursuit of Katherine is in all their best interest. Tranio encourages all the others to eat and drink with them as friends, since they share the common goal of having Katherine marry Petruchio.
As Bianca's suitors cooperate as friends in their common goal of getting Katherine married, their pursuit of Bianca seems almost as important for establishing their bonds of friendship as for the actual desired marriage to Bianca.