The main plot revolves around the marriage of Kate, the shrew of the title. She is fiery, says exactly what she thinks without a thought and can be insulting to those she has no respect for. She has no wish to marry - and even if she did no-one is prepared to take on a sharp-tongued, bitter woman who doesn't know her place. She is made to seem even worse in the movie as she physically assaults her sweet tempered, younger sister, Bianca, after tying her hands. But her father has said Bianca cannot marry until Kate does so the pressure is on to find the older sister a husband no matter what she thinks - and there's a hefty dowry to encourage suitors.
Enter Petruchio, a thuggish, money seeking adventurer - the first time we meet him he is assaulting his servant for not acting on his confusing demands - in search of a wealthy wife. Kate's dowry attracts him immediately. He starts his courtship by completely ignoring whatever she says and claiming any abuse is affection and when he makes an offer her father accepts with alacrity and Kate is forcibly married off. In the movie she's cut off as she tries to say no which makes what follows even more disturbing. From then on things go even more badly for Kate. Petruchio makes it clear that she is now no more than a possession and she's bullied, deprived of food and sleep, left to wear rags, humiliated and abused to force her into submission. Finally she submits to whatever outrageous demands - and there are many of them - that Petruchio makes of her.
The sub plot is a little less distasteful as Bianca - the sweet natured, obedient daughter whose father is basically arranging to marry her to the highest bidder - runs off and marries her own choice but her marriage to the son of a wealthy man (after a number of mix ups) is accepted, when they seek her father's forgiveness. At Bianca's wedding reception Kate's obedience is tested by Petruchio and she obeys him immediately and then berates the other women for not doing as their husbands demand. In some productions - Zeffirelli's is one - her final speech is subverted as she winks at the end implying she is complying but not 'tamed' but, for me, this doesn't fit with the rest of her behaviour. This is a woman who has no other option than to submit.
So what does the play tell us with an abusive and misogynistic husband is being lauded as admirable and a submissive wife as an ideal? I think it leaves an unpleasant taste behind and apparently this isn't just a modern, feminist idea. Even at the time there were those who found the subject distasteful including the playwright, John Fletcher, a contemporary of William Shakespeare, who wrote The Woman's Prize or The Tamer Tamed in which Petruchio, now older and widowed, remarries Maria, a shrewish woman, and gets his comeuppance.
One school of thought is that Shakespeare was writing a satire and the play is awash with irony and another that this is a simple farce. If either was his intention I don't think he succeeded. Petruchio is not in any way an admirable or amusing character - he's brutal, and Kate is not the only one on the receiving end of his brutality - and, while Kate is certainly a difficult person, there is, for me, no humour in the way she is treated.
For all that The Taming of the Shrew remains popular, being performed frequently and even forming the basis for the popular Cole Porter musical Kiss Me, Kate.