There's a lot to examine regarding relations between the genders in Shakespeare plays. Here's a brief overview of a few examples that you could choose to analyze further.
In Macbeth, Lady Macbeth subverts gender roles. She seems to dominate the marriage, embarrassed by her husband's reluctance in the murder of Duncan. She questions his masculinity. Meanwhile, she calls upon spirits to "unsex" her so she can do what is necessary to overthrow the King. Additionally, we could look at how the three witches influence Macbeth. These weird women lead to his downfall and essentially the actions of the play, making us question if women have more influence than men.
In A Midsummer Night's Dream, Oberon and Titania are at odds with each other. They argue over a changeling boy in Titania's care and the "forgeries of jealousy" have caused an imbalance in nature. We can also look at Hermia's love triangle: she is in love with Lysander, but her father wants her to marry her suitor Demetrius. As a woman, Hermia is not able to decide for herself who she can marry. Her father chooses, and the only choice the Duke gives her is to listen to her father, join a convent, or be put to death.
In Measure for Measure, Angelo imprisons Isabella's brother. He says Claudio will die unless she gives herself sexually to him. Isabella says she will tell everyone what Angelo has said, but he laughs, reminding her that the public would believe him over her. Men have more power, and society is more likely to believe them. With the help of the disguised Duke and Angelo's former betrothed, Isabella is able to trick Angelo, giving her some power back.