This is a huge question, so I can only give you a brief overview. Maybe you can break this up and ask more specific questions about specific plays if you need more information.
A Midsummer's Night Dream
Titania, Queen of the Fairies, battles with her husband, Oberon, the King of the Fairies, over who gets to keep a little Indian boy whose mother has died. Titania wants him because she promised the boy's mother she would care for him, so she goes toe-to-toe with Oberon, who likes to have his own way. This leads to bad weather in the human world—and comedy in the fairy forest.
Hermia battles her father and Theseus over the right to marry for love. Theseus literally captures his wife-to-be in battle.
The Taming of the Shrew
Katherine, the "shrew," is wed to Petruchio, who marries and "tames" her by using his power to deny her food and drink, saying what she is served is not good enough for her. It's significant that this is presented as a play within a play, suggesting that Shakespeare thought it wildly improbable that such a taming could be anything but a fiction.
Lady Macbeth wages psychological battle with Macbeth to compel him to murder Duncan and take the throne.
Lear battles his evil two oldest daughters, who lie and swear undying love to him and then cast him out once they have his kingdom and power.
Measure for Measure
Women do battle with the evil Angelo and win. Sex with the pure Isabella is Angelo's price for commuting her brother's death sentence. Angelo has recently broken his engagement with Mariana because she has lost her dowry. Isabella and Mariana hatch a plot: Isabella will agree to go to bed with Angelo, but only in complete darkness. Then Mariana substitutes herself for Isabella. Once he has consummated the relationship, Angelo is forced to honor his engagement and marry Mariana.