In the Taming of the Shrew, Petruchio recognizes, respects and desires Kate's intelligence and strength of character. He does not want to conquer or truly tame her. He is a man who is very confident in himself and does not want or need someone to massage his ego. Petruchio seems to me to be a man of sport and challenge and likes to surround himself with witty, challenging people. He wants in a mate what Kate has - fire.
From Petruchio's response to his friend Hortensio (I.ii.64-75), it might be said that Petruchio came to Padua to make himself richer by marriage, to any woman, no matter how wretched. Petruchio is not in desperate need of money (I.ii.56-57). He tells Hortensio (I.ii.49-57) that his father has died and that he is out in the world to gain experiences he cannot at home and only secondarily to find a wife. Also, immediately before this declaration, is the scene of misunderstanding between he and his servant Grumio about knocking on the gate (I.ii.5-43). I see this exchange as demonstration of his enjoyment of verbal sport, a good example of Petruchio's sense of humor and his appreciation of things non-conventional. Though Petruchio may not agree with what society has determined to be proper and dignified, he is aware of the importance of appearing to conform. In what he says to Hortensio, I feel he is simply extending this sport and humor into the ironic.
It is in Hortensio's description of Kate that I believe Petruchio's interest is captured. Hortensio describes Kate (I.ii.85-89) as wealthy, young, beautiful, properly brought up intolerably cursed, shrewed and froward. Though Hortensio finds the last three traits negative characteristics, Petruchio appears to be a man who also posses, and is proud of, these negative qualities. That the qualities are considered negative in Kate and not Petruchio is a reflection of the societal standards of the fifteen hundreds. It was okay for a man to be that way, but not a woman. Petruchio is the kind of man who would want a mate with similar qualities to his own to challenge him, sharpen his wits and keep his interest. If he had wanted someone who was conformed to societies expectations, or who had already determined to deceive by concealing opinions and views, he would have chosen someone more like Bianca. However, Petruchio is a clever man who sees beyond faÃ§ades because he uses them, in addition to a lot of irony himself (II.i.46), (II.i.283-289).
It is clear in Grumio and his other servants (as demonstrated in the opening of act 4 (IV.i.1-113) that Petruchio prefers the interesting to the conventional. But because Petruchio understands the ways of society, he knows he must demonstrate to Kate the importance of proper public appearance. To Petruchio it is appearance rather than genuine conformance that is important. Otherwise, the woman he loves would be called names and treated in ways Petruchio might be required by honor to defend.
In his ironic way, Petruchio does speak consistently about making Kate yield to him (II.i.124,136), (II.i.269-271) and of his monetary motivation (II.i.123,124). But, his methods are sportsman-like (Falconry, (IV.i.183-190) and game-like demonstrations of the outrageous (beating Grumio because Kate's horse stumbled IV.i,68-80). Petruchio's servants like him very well and enjoy his entertainments. In what Petruchio says following he and Kate's first meeting (when her father walks in with Gremio and Tranio (II.i.269)) it becomes clear just how heavily Petruchio employs irony. He states that he is born to tame and conform Kate. Though the servants he has chosen to surround himself with are neither tame nor conforming to what most would consider proper servants. He also says he must and will have Katherine for his wife. This is a man who is completely taken by this woman: he called her properly by her formal name and says he will have her. Petruchio is as taken by Kate's person as the other suitors are taken by Bianca's beauty and coyness.
In the above scene, Petruchio tells Kate to never make denial. He knows she is not yet convinced, but is telling her to trust him and go along with what he says for the sake of appearance. This slowly sinks into Kate and finally takes hold when she understands Petruchio's way of irony on the way home to her father's (IV.v.12-22). Because they are so much alike, Kate takes very quickly to Petruchio's games of words and irony (IV.v.37-50). Petruchio is the kind of lively person who would be disappointed in a victory too easily won, and disappointed in Kate if she were genuinely tamed. I feel certain she will have her victories, and Petruchio will enjoy them as much as his own.