A good place to start when trying to establish points of connection between us and the plays of Shakespeare is trying to put yourself in the shoes of some of the characters and the position that they face. When I first teach this play to students, I start off by looking at Act I scene 1 when Egeus enters with his daughter and Lysander and Demetrius. I normally start off by asking my students whether they think they should have the right to pick the person they marry or whether their parents should pick their future life partner. Normally, the overwhelming response is that they believe they have the right to select who they will marry! Then I ask them how they would respond or feel if their parents tried to force them to marry somebody that they had chosen for them rather than their own personal choice. This evokes a range of emotions, which prepares them for reading about the plight that Hermia finds herself in and automatically establishes a connection between the plot of the play and my students. In particular we look at Egeus's lines when he says:
I beg the ancient privilege of Athens:
As she is mine, I may dispose of her,
Which shall be either to this gentleman
Or to her death, according to our law
Immediately provided in that case.
A close examination of his speech looks at how he talks and the kinds of language he uses to talk about his daughter, for example talking about "disposing" of her, which again leads back into the discussion of how Hermia feels and how would they feel in the same situation. Activities such as this give students a real interest in the play and help them to sympathise with the issues and characters contained therein. Hope this helps!