So mainly this story is about a guy who makes a bet with some other guys about seeing if he can win Katherina over?
So is the whole story about this girl who could not be tamed until she falls in love with this guy who made a bet with some other guys?
Not at all. The Taming of the Shrew is about relationships--not just Kate and Petruchio's, but also others. Through Kate's resistance, and Petruchio's wild demands, we see our own battles in our own relationships. It also makes us question where power belongs in relationship, as well as where it truly lies. Petruchio does not make Kate love him--he makes her obey. Whether she does so out of love or necessity is open for interpretation, but we watch her shift. We sense her frustration, and, depending on how we read her motive, we may also see her sense of humour developing, replacing the anger, and controlling her actions; or, we may see her giving in to a power greater than her own. Either way, this play leaves us thinking about power, right action, and choices. One thing is certain, anger, even anger attached to a brilliant wit, doesn't have the power we sometimes think it does. Anger is loud, but it is not particularly effective.
To answer your second question, this whole story is about a girl who could not be tamed until she meets a guy who really doesn't care if she's tamed. Petruchio wasn't originally interested in Kate's love or obedience; he wanted the deep dowry the Baptista was offered. Most of why Petruchio was able to tame Katarina was simply because he wasn't all the interested in what she thought of him.
Not exactly. The bet you're discussing actually happens later in the play, once several various plots and sub-plots are resolved, we're left with 3 couples. One of them is Kate and Petruchio, Kate being the "Shrew" of the title (a domineering woman).
Petruchio makes a bet that his wife is the most obedient, so all three men call their wives to their side. Only Kate, the former shrew, comes, and Petruchio wins the bet, having "tamed his shrew".
Not really. This book "Taming of the Shrew" mainly focuses on the themes of marriage and courtship and also dealt a great importance of the married life after the wedding. It is mostly talking about relationships with other people and how they impact their daily lives.