"I'll not budge an inch, ..."See in text(Induction - Scene 1)
This now common saying became popular with this play. "Budge" was either not a verb, or not a commonly used verb before Shakespeare used it here to mean "stir or move." Sly represents a drunken playgoer who London's magistrates often complained about. Sly indulges in all of the corruptions associated with the theater and illicit entertainment outside the city walls, such as bear baiting and whorehouses. In this way, Shakespeare begins his play making fun of some of his audience members by including this character.
"Come, madam wife, sit by my side
and let the world slip: we shall ne'er be younger...."See in text(Induction - Scene 2)
"Let the world slip" is Shakespeare's take on the then popular phrase "let the world wag." Sly's phase soon became the popular saying as it is now. Sly's introduction to the play as a way to pass the time away is a meta way to comment on the very thing that the audience is doing by sitting in the theater and watching this play.
"Please ye we may contrive this afternoon,
And quaff carouses to our mistress' health,
And do as adversaries do in law,
Strive mightily, but eat and drink as friends...."See in text(Act I - Scene 2)
Tranio's lines underscore a major theme and problem within the play. Love is devoid of meaning and feeling in a world where suitors compete with each other for money, status, and marriage. Feeling is displaced and love becomes more of a contract or prize that can be likened to legal battles. The play will continue to explore this problem and, ironically, offer Katherina and Petruchio's relationship as a solution at the end of the play.
"As are the swelling Adriatic seas:
I come to wive it wealthily in Padua;
If wealthily, then happily in Padua.
..."See in text(Act I - Scene 2)
Petruchio sets up his motives within these lines. He has come to Padua to marry a rich wife regardless of her disposition, physicality, or internal makeup. Petruchio claims to not care about love and only care about money. In this sense, Petruchio establishes himself as a character that needs to be "tamed" just as much as Katherina does: while she needs to learn kindness, Petruchio needs to learn affection.
"KAT:
Asses are made to bear, and so are you.
PET:
Women are made to bear, and so are you.
..."See in text(Act II)
This first exchange between Petruchio and Katharina shows that their wits are matched. While Petruchio believed that convincing Katharina would be easy, and Katharina believed that getting rid of Petruchio would be easy, both realize that they were wrong during this exchange. Notice that their quick back and forth lines come across like a duel of words, each turning the other's words or metaphor to use against the other.