For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Act I
The first act begins with Dionysus declaring that he's in the midst of punishing the house of Cadmus for denying that he's a god. He's taken over the minds of the women of Thebes and sent them off to the woods to dance in celebration of him. We know there's going to be serious trouble when King Pentheus returns and swears to squash this Dionysian uprising.
Act II
The conflict between Pentheus and Dionysus continues to drive the play. Some of Pentheus's men try to grab his mother Agave from the rituals. This causes her and the other Maenads to ravage the countryside. Meanwhile Pentheus tries to imprison the Stranger, who is really Dionysus in disguise. The act peaks when the god summons earthquakes and lightning, squashing the stable where the King tried to imprison him in.
Act III
The final act is propelled forward when Dionysus convinces Pentheus that he should check out the Maenads rituals for himself. The King dresses as a woman to disguise himself. This does him no good at all and he is ripped apart by his own mother. The play peaks as maddened Agave returns bearing her son's head and slowly realizes exactly what she's done. Euripides wraps the play up with Dionysus dealing out punishments to Agave and Cadmus.