In Cat's Cradle, John gives us all the info we need to understand the book thanks to a very convenient story-telling device: he's writing a book.
Yep, John is writing a book on the atomic bomb. To do this, he goes around gathering the research necessary by writing letters to Newt and interviewing the people of Ilium. Conveniently, he also provides the reader with all the research (read: exposition) they'll need to understand the story. Also, the way John reacts to the characters and information tells us about him as a character as well.
Act II begins when John shows up on San Lorenzo, and it revolves around what's called "the rising action." Basically, that means the protagonist is in conflict with the antagonist and can't quite figure out how to win.
Cat's Cradle's second act can be kind of hard to pin down, for various reasons. One, the closest thing the novel has to an antagonist, Felix Hoenikker, is good and dead although his surrogate, ice-nine, is alive and well on the island.
Also, John is unaware that ice-nine exists, and his ignorance makes the conflict harder to spot than in, say, Star Wars where the people shooting at each other are obviously in conflict.
But the conflict does exist, and every step John takes on San Lorenzo puts him directly into conflict with ice-nine. The way he befriends the Hoenikkers and the island inhabitants, how he becomes President of the island, his desire for Mona, and the plans he makes for improving the island life: all of this brings him closer and closer to realizing ice-nine's existence even if he's unaware of it the whole time.
Act II ends with what's called the climax—the moment everything changes for the characters. And that, dear Shmoopers, is when ice-nine is released into the ocean, effectively ending the world.
Doesn't get more climactic than that.
Nowhere else for Act III to go now. It has to provide the resolution.
In Cat's Cradle, that basically means showing how the survivors deal with their new world. Some, like Mona, commit suicide. Philip and Julian Castle die trying to save others. And John joins a small band of the survivors who do what they can to, well, survive. The novel reaches its final resolution when John meets Bokonon, and the two discuss what to do with the world in such a state.
Of course, if you're looking for something with a little more spoilers, there's always our "What's Up with the Ending?" section.