Act I of The Moon is a Harsh Mistress corresponds to Book 1. This section gives us the story's exposition. We discover who the characters are, the future they live in, and the conflict they'll undertake. We learn about Mannie and Mike and how their relationship works, as well as about the Luna customs and how the Lunar Authority has created a situation that will eventually lead to a lack of resources and then, naturally, cannibalism. Finally we see Mike, Prof, Wyoh, and Mannie planning a revolution against the Authority because, again, cannibalism.
Act II corresponds with Book 2, presenting us with the rising action of the story. All those plans we saw being crafted during Act I go into effect and the conflict heats up. With the Warden and Security Chief overthrown, Prof and Mannie head to Earth to convince the people of the Federated Nations to accept the legitimacy of the Luna free state. Against the Federated Nations' unwillingness to accept them, the two face odds they can't overcome and ultimately fail. Luckily, they escape Earth, and Mannie learns their failure was part of the plan the entire time.
Book 3 presents us with Act III, the wrap up and resolution. During this section, the Federated Nations attack Luna, but the Loonies defend themselves. Mannie then begins the operation to bombard Earth with moon rocks from the secretly built catapult, Little David's Sling. There's an intense standoff as Mannie and Mike attempt to defend the Luna and the catapult from the Federated Nations' warships, but just before they run out of ammunition, the nations begin recognizing the Luna Free State. The revolution is a success.