Power and control in marriage
Every marriage in the play involves a negotiation over who is in charge. Petruchio makes that negotiation explicit and aggressive. Even the Bianca subplot, which looks gentler, ends with Bianca refusing to obey. The play maps out different versions of marital power without settling on one as correct.
Performance and identity
Characters constantly pretend to be someone else. Lucentio becomes a tutor. Tranio becomes Lucentio. A merchant becomes Vincentio. Petruchio performs the role of devoted husband while starving his wife. The Sly frame makes the whole play a performance. Identity in this world is something you put on.
Gender and social expectation
Katherina is called a shrew because she speaks her mind and refuses to be managed. The play shows how that label is used to pressure her into conformity. Whether she actually changes or just learns to hide her resistance is the question the play refuses to answer cleanly.
Deception and disguise
The Bianca subplot runs entirely on disguise. Almost every character lies about who they are at some point. The play treats deception as a normal tool for getting what you want, which makes it hard to trust anyone's sincerity, including Katherina's final speech.
Wealth and marriage as transaction
Petruchio says outright that he came to Padua to find a rich wife. Baptista sets conditions before allowing courtship. Dowries are negotiated like contracts. The play does not hide the economic logic behind the romantic one; it puts them side by side.