The play is called The Taming of the Shrew, and it is undoubtedly the "Shrew" story which most captures the attention of audience or reader; the alternative plot of Bianca's suitors and Lucentio's misrepresentations being dull and conventional by comparison. The two characters of the 'shrew' drama are fascinating and enjoyable to watch.
The most important player in the drama is not the 'shrew' herself but the man who tames her, who indeed has a great deal more to say in the play than Katherina herself.
It would not be unreasonable to describe Petruchio as a swaggering, machismo-filled egotist. He is supremely self-confident, he is entirely selfish in his desire for a rich wife and a conformable one, and ruthless in his intention to 'tame' her by whatever means he deems necessary. Yet somehow, Petruchio manages to be likeable. Perhaps it is because, despite his cunning and play-acting, he is more honest than those around him, and more perceptive. He does, after all, perceive Katherina as a wife worth having, and worth going to a great deal of trouble over!
His first appearance shows Petruchio as a bluff character who expects immediate obedience—he wrings Grumio by the ears for not doing as he is told. Yet there is also the clear suggestion that he is not such a hard master as all that, since his servant is quite willing to "misinterpret" and indulge in a bit of word-play. A truly harsh master would not have such a cheeky servitor, but Petruchio participates in the game, and turns the tables by changing 'knock' into 'ring' and wringing Grumio's ears. It rather looks as though he enjoys the game.
Still, Petruchio is a practical man. In the speech to Hortensio by which he introduces himself to the audience, he states concisely that he seeks to increase his wealth by the acquisition of a rich wife. (This contrasts pleasingly with Lucentio's rather vain pronouncements about his own reasons for being in Padua.) In a most businesslike fashion he goes on to settle the matter of Katherina's dowry before he even sets eyes on her. Provided the condition of wealth is met, he avers that the woman's other qualities are of no interest to him. This is unattractive, but is ameliorated by subsequent revelations to the effect that other things do seem to matter to Petruchio after all.
Certainly he is not a bit deterred by the description of Baptista's daughter as a shrew. As Petruchio, the man of action, points out, he has heard worse things than a woman's tongue—stormy seas, gunfire, battle, etc—and he is plainly not the slightest bit impressed by the tales the lesser men, the scholars, tell of Katherina. Moreover, Grumio—who is presumably in a position to know—does not consider that a woman's scolding will have any effect on Petruchio.
Petruchio is utterly self-confident, and a man who has survived actual perils, certainly not one to be intimidated by a 'curst' female. In fact, it appears that the opposite is true, and he actually rather likes the idea of a 'proud-minded' woman. He is obviously entertained by Katherina's breaking the lute over Hortensio's head ("Now, by the world, it is a lusty wench"). He himself has already shown impatience with his friend's and others' over-wordy pronouncements ("Hortensio, to what end are all these words?"). Moreover, at the end of the play, he takes distinct pride in his wife's ability to deal wittily and scathingly with the Widow's unkindness, he wagers that Kate will come off with the honours, and positively encourages her to have a go at the other wives. The sweet, submissive woman that Bianca appears to be would not suit him nearly so well.
By the time Petruchio meets Katherina, he has already settled the vital matter of her dowry, heard about her bad temper, and devised a means of dealing with her. His tactics demonstrate his cleverness and his effectiveness. He does, after all, get his way. His behaviour at their first meeting, in contradicting her, seems perverse, but although his contradictions irritate and confuse Katherina, he also gives her much that she could reasonably be expected to want. From the very beginning of their relationship, he presents her with a revised version of herself, a desirable and praiseworthy woman. (And he continues to praise her and treat her as a cherished lady even as he prevents her from eating, sleeping or acquiring new clothes.) He also demonstrates that he is capable of matching her wits in word-play, and he is quick and cunning enough to cut off her avenues of escape, by pre-empting them.
Later, of course, he humiliates her, appearing late for the wedding in his absurd clothes, and behaving outrageously. Public humiliation was a traditional method of shaming a scolding woman, in the 'skimmington' which would mock her in front of her community. Petruchio, a magnificently flamboyant character, can carry off his absurdities quite brazenly, but by appearing in such a guise he is to a great extent sharing the embarrassment of the situation with his bride (and, perhaps, shaming the father who could not discipline his own daughter properly). Petruchio is not particularly interested in outward grandeur, so it is no great penance for him not to behave and dress as a great lord should. He does not apologise for himself, but expects to be taken at his own value. Towards the end of the play, in the encounter with Vincentio, he makes no attempt to apologise for his own extraordinary rudeness, but ignores it completely when he has made his point and returned to normal behaviour.
Petruchio's methods for taming his shrew vary as time goes on: next he deprives her of food and sleep, a technique he compares to the methods used to tame hunting hawks. Though the modern reader may dislike the equation of wife with animal, the contemporary audience would have understood the value of a tamed falcon. Petruchio obviously regards Katherina as something precious, and worth the considerable trouble of taming. Indeed, when contrasted with other possibilities such as the vicious scold's bridle, or plain, simple beatings, Petruchio's treatment of his bride seems comparatively gentle.
There is also the fact that he is obviously strongly attracted to Katherina when they meet. During the cut and thrust of their word-play, he cannot help introducing sexual innuendo into the dialogue, showing which way his thoughts are tending! He describes her as a beauty, calling her "the prettiest Kate in Christendom"—something she has undoubtedly never heard before—and comparing her lovingly to the hazel-twig, "straight and slender" and "sweeter than the kernels". His compliments are attractive and unusual, original as befits a man of wit, not culled from oft-used sources. And he is obviously determined that she shall be his wife, not just for her gold, but for her self as well. From this first meeting onward he repeatedly requires her to kiss him, which at the very end of the play is swiftly followed by his taking her off to bed.
Macho and blunt he may be, Petruchio's cleverness, wit and strength of character show him to be a worthy mate for Katherina. He is 'up to her weight' and can deal with her anger rather than resenting it and belittling her as the other men do. He is worthy of her respect, and is able to give her the respect she craves.
What of Katherina? Long before she actually appears, we know she has a reputation for being 'curst', and she is certainly wilful and contemptuous of those around her. Her contempt is not altogether unreasonable, since clearly no-one is capable of matching her native wit until she encounters Petruchio. Katherina's wit is splendidly quick, and she can deal double meanings far more cleverly than her feeble opponents. Her very first speech:
"I pray you, sir, is it your will
To make a stale of me among these mates?"
is beyond Hortensio's ability to counter, and her battle of words with Petruchio shows that she has a very sharp mind indeed. She treats those around her with contempt because she sees nothing to respect in them, and because they will not accord her respect, she gives them cause to fear her.
There is a great deal of anger in Katherina, and again, it does appear that she often has reason to be enraged. The obvious preference of everyone around her for her younger sister is the source of much of it, as she clearly resents her father's favouritism and her own humiliation.
"She is your treasure, she must have a husband,
I must dance bare-foot on her wedding day..."
On her own disastrous wedding day, Katherina is full of resentful I told you so's—to which nobody pays any attention. She is a proud girl with a strong sense of her own worth; the lines
"I see a woman may be made a fool
If she had not a spirit to resist"
seem to embody her sense of worth and her determination not to be humiliated.
Of course, as a young woman, she has no other power than that of her own personality. Her father effectively owns her and can dispose of her as he pleases, and meanwhile as an unmarried, uncourted girl she is very low in the social hierarchy. For a woman with a brain and strength of mind, the restrictions of her situation must chafe extremely, and Katherina is not a personality who can play along with the system in order to get what she wants (as Bianca does). She may be a shrew, but it is because she is too honest to be otherwise.
Yet it is clear that Katherina behaves 'shrewishly' in reaction to her circumstances—Petruchio says "it is for policy", but it is less premeditated than that—for when her circumstances change she is perfectly able to behave reasonably. When her new husband is outrageous and unfair, she attempts to calm the situation. There is more to Kate than a bad temper.
Either one of these characters is enough to bring vibrant life to a scene, whether it is Petruchio brangling with his servant at the gate or Katherina shrieking at her sister. However, when the two are together, there are real sparks! The first scene in which they meet is a magnificent battle of wits, with both playing with language in a quite dazzling way, so that their duel becomes almost physical. Even before Kate strikes Petruchio, it is possible to imagine them lunging to and fro across the stage as though wielding foils. It is also a very sexy scene. In their wit, these two are reminiscent of Much Ado About Nothing's Beatrice and Benedick, who enjoy similar battles of words—although Beatrice and Benedick are equals in a way that Katherina and Petruchio are not. Katherina has no worthwhile status except what she can win with her 'curst' tongue until Petruchio elevates her by marrying and transforming her. Still, by the end of the play, Petruchio and Katherina have become a team, and woe betide anyone who opposes them!
The other characters in the play are nowhere near as interesting to watch. Bianca, apparently presented as the ideal of womanliness, is dull next to her fiery sister. Being a model female, modest, quiet, and quite unthreatening to the egos around her, she does not have a great deal of scope to be interesting. There are, of course, hints that she is not altogether as biddable as she is thought to be: she is happy to connive at Lucentio's illicit courtship and to flirt with him even though her father has expressly forbidden her suitors until her elder sister is married. Bianca insists also that she will "not be tied to hours nor 'pointed time, But learn my lessons as I please myself". She is willing to do as her father tells her—at any rate, while he is there to watch her—but also perfectly ready to wield authority of her own. There is also, perhaps, a sly dig at Katherina, when Bianca tells her:
..."what you will command me will I do
So well I know my duty to my elders."
Perhaps Bianca is taking the opportunity to remind her sister that while Bianca is young and desirable, Katherina does not seem to be in the same category.
So at the end of the play it is not particularly surprising that Bianca turns out not to be quite such a soft, obedient wife as Lucentio had assumed he was getting. He was more than happy to take advantage of her disobedience in order to woo her, but unlike Petruchio, was misled by his chosen woman's public behaviour.
Lucentio's courting has nothing of the fieriness of Petruchio's. In his expository speech at the beginning of the play, he has a great deal to say about how he has come to seek "a course of learning and ingenious studies", and immediately makes plans for the entertainment of his friends. Later, he approves Bianca's statement that she will go to occupy herself with her books and instruments, suggesting that he values these things... but he does not himself embark upon a course of study at all, and all his learning is fit for is to seduce Bianca. His self-deception and pious public face are a complete contrast to Petruchio's blunt honesty. The wooing Lucentio achieves under the cover of learning is dull stuff compared to the lightning wit Petruchio has to employ to keep up with Katherina.
Bianca's unsuccessful suitors are also an uninspiring lot. Hortensio has quite a bit to do to further the plot: he tells Petruchio about Katherina, he acts as a foil to Lucentio the favoured suitor, and he provides the third husband necessary to give proper dramatic balance to the final scene. But his different functions do not seem to add up to a coherent character, and it is very difficult to see in this personality-free creature the "best beloved and approved friend" of Petruchio, whom we would expect to have higher standards.
Gremio is rather a stock figure of fun, an old man eager to marry a young woman. Although his romantic pretensions are made fun of, he does have some enjoyable speeches, particularly his description of Katherina and Petruchio's wedding.
Tranio, who assumes the role of suitor, is the manipulator and the cunning servant who is so useful to many would-be heroes, but not much else. The best that can be said of him is that he is willing to lie, cheat and deceive in the service of his master.
The girls' father, Baptista, is obviously a conventional figure of the old school, wishing his daughters to be properly married and considering it his business, and none of theirs, whom they marry. Although he does mention the notions that their husbands must have the girls' love, in practice it is not a relevant matter: he gives Kate to Petruchio in spite of her very explicit refusals, and he assures the two bidding suitors for Bianca's hand that whichever makes the best offer shall have her 'love'.
Baptista is easily duped, by Tranio, by Lucentio, by Hortensio, even by Bianca, who is not quite the paragon he thinks her. He is also perfectly willing to accept Petruchio's assurances that he has won Katherina's heart, despite the fact that Katherina angrily and vehemently denies this—so either Baptista is capable of great self-deception, or else his desire to get his elder daughter out of the house is so strong as to overwhelm any wish he might have of ensuring her happiness. Or possibly both. With the main characteristics of gullibility and very indifferent fatherhood, he does not really come to life, and the scene in which he auctions Bianca to the highest bidder is probably the most tedious in the entire play.
Two other characters in the play do seem to come alive, however. One is Grumio, the confusingly-named servant to Petruchio. He is obviously a very impertinent attendant, in his first scene wilfully arguing and being obstructive, presumably for his own amusement. He carries on a kind of commentary during the scene in which Petruchio and Hortensio discuss Katherina, and displays distinct wit in the process, even though he is more interested in his stomach than in anything else! Later, he recounts the married couple's sorry progress towards Petruchio's home, and again his word-play is in evidence, possibly on a more knockabout level than his master's. He is even able to stand up to Katherina, continuing Petruchio's regimen of denying her food under the guise of taking special care of her. Grumio is a cunning servant just as Tranio is, but although he does not further the action of the plot, but merely comments on it, he is much more interesting than Tranio.
The other character who comes alive is the drunkard who is taken up by the hunting lord and led to believe he is a nobleman, in the Induction to the piece. Although this predicament is a well-known comic scenario, Shakespeare seems to have brought Christopher Sly to life in remarkably few words. Sly is resoundingly honest about his own status when he wakes up, but rapidly adapts to the elevated situation in which he finds himself—there is a hint right at the beginning that he would not be unfertile ground for such a trick, when he refers to the 'Chronicles' and some befuddled idea of an ancient lineage for his family. He is quick to take an interest in the 'wife' presented to him, and obviously disappointed not to be able to whisk 'her' off to bed at once! He attempts to live up to the lordly image the tricksters present him with, but is unfortunately unable to stop himself nodding off while watching the play... I felt quite sorry for Sly, and could not help wondering how he would feel at the end of the play, sent back to his proper station in life. It is quite disappointing that his story fades into oblivion.
However, Sly and Grumio are not major players in the drama as a whole, and although they enliven the action, being believable and interesting to watch, there is not the substance to either that there is to Petruchio and Katherina.
This is certainly also true of the other characters in the play, even though Bianca and Lucentio ought to be very nearly as important as Petruchio and Katherina. But the courtship of Bianca is formulaic, and the manipulations of the disguised Tranio and Lucentio's fathers both fake and real, are frankly tedious. These characters are essentially following the steps of a formal dance; Shakespeare has endowed them with some style, but they are well-known 'types' treading a prearranged path—Lucentio is the clever young man destined to get the girl by trickery, Baptista is the gullible authority figure who will be outwitted, and so forth. They therefore engage the reader's interest not for themselves but only for their places in the plot—of course, an audience has the benefit of seeing them breathe! Katherina and Petruchio seem to be performing a far more dashing tango. These two are not so easily defined (despite such labels as 'shrew' and 'tamer'), and although it is clear that in the end the 'shrew' will be tamed, there is real excitement in the process. In short, I have to agree that Katherina and Petruchio are indeed the only true characters in the play.