How does the setting influence events and characters in The Taming of the Shrew?
I'm not sure the setting influences the characters. In other words, it seems unlikely to me that Shakespeare (or any playwright, for that matter) would have created these settings and then decided how they will impact the characters. Instead, let's think about how the characters' actions fit (are suited or appropriate for) the settings.
At Senor Baptista's house, the dynamic is clearly dominated by Katherine. She is tyrannical and in charge, and her father and sister clearly live in fear of her. When Petruchio enters the house, the balance of power shifts, but Baptista and Bianca still have no control.
At the church, Katherine is no longer in control of her circumstances, nor is Petruchio. This seems to be Baptista's domain, to some degree, as Katherine is subdued for once and he thinks he's doing what she wishes--or at least he's convinced himself of that.
The journey is a setting unto itself, full of mud and rain and whatever other elements could be gathered to conspire against Kate. Though we only hear about this setting, we can appreciate the fact that even the elements are attempting to tame the shrew.
Petruchio's house is a disaster. It's a poor man's bachelor pad, suffering from the neglect of both poverty and lazy servants. This is clearly Petruchio's domain...but not for long. As Kate figures out his "taming" strategy and begins to play along, she also wins the hearts of the servants and makes his home a place where she can live and love.
The journey back to Padua is also a setting; once it worked against Katherine, but this time she mastered both the sun and the moon, so to speak.
Finally, they return to Kate's father's home. It has now become a more neutral setting, suitable for the wagering done in the final scene.
I agree with auntlori, I'm not sure that you can identify how the setting influences the characters. You would need to actually see the characters in at least two different setting choices in order to see how any setting influences them. Not possible from just reading the play.
You could, however compare staged and filmed productions to notice different settings and see the effect that they have on you, the audience. You can also see what different aspects of each character are brought forth when you compare productions set in different places and times.
It should be remembered, when discussing the setting of a play, that there is nothing about the place and time that is set in stone. Unlike a novel, the text of a play is simply the words spoken by the characters. A setting described as Padua, for example can easily become, as in the movie and TV show 10 Things I Hate About You, Padua High School. This TV show isn't intended as a production of The Taming of the Shrew, merely inspired by it, but the setting choice of a 21st century high school could still work for a staging of the play.
So, it's the effect of the setting on the audience you might really be interested in. And potentially, as I mentioned above, comparing different productions with different settings in order to compare how the characters behaviour seems influenced by the settings.