Lysander’s plan is to run away with Hermia to his aunt’s house, where they can be married beyond the reach of Theseus and Egeus.
Hermia wants to marry Lysander, and he wants to marry her. Hermia’s father Egeus wants her to marry Demetrius, and he hauls her before the Duke of Athens so he can force her to marry him. Hermia is upset, but Lysander has a plan.
I have a widow aunt, a dowager
Of great revenue, and she hath no child:(160)
From Athens is her house remote seven leagues; ...
There, gentle Hermia, may I marry thee;
And to that place the sharp Athenian law
Cannot pursue us. (Act 1, Scene 1)
They decide to sneak away at night, and go through the woods to Lysander’s aunt’s house. Since she considers Lysander like a son, she will help him. Once there, they will be out of Theseus and Egeus’s reach. They can marry, and live happily ever after.
The plan might have worked, except that Hermia told Helena, and Helena told Demetrius. Helena wanted to marry Demetrius, and was angry at Hermia. The next thing you know all four of them are chasing each other through the woods. Eventually, they each get married to the right person.
Lysander's plan is to run away with Hermia. He tried winning Hermia's hand in marriage by impressing Egeus. But Egeus has his heart set on Demetrius marrying his daughter. Realizing that Lysander, poorer of status, has no hope, he plans to elope with Hermia to his aunt's house. His aunt has some money.