Here's how things go: Manjiro, a teenaged Japanese fisherman, and his pals get caught in a storm at sea, and wind up stranded on a deserted island. Luckily, though, they get saved by an American ship—the John Howland—which is led by the good Captain Whitfield.
Manjiro and the captain hit it off so well that Manjiro goes to live in America (Connecticut specifically) with the captain. There, they become a family that eventually grows to include a new Mrs. Whitfield and a baby, William Henry.
They all live and work on a farm together, and Manjiro basically turns into an American teenaged boy, complete with school, a couple of guy friends, a girl he's into, and a class bully.
But he still misses Japan and his family there, so when he gets the offer to return on a whaling ship with an old friend from the John Howland, he takes it. Only his friend, Ira Davis, who's been made captain of that ship, turns into a crazy tyrant. The crew engages in mutiny and things turn out okay, with Manjiro eventually getting promoted to harpooner—but he never makes it back to Japan.
Instead he returns to Connecticut, where he finds out that his baby brother died. He wants to stay with the Whitfields, but his friend Terry convinces to take part in the Gold Rush in San Francisco. They go, and Manjiro strikes gold—and with this gold, he goes to Oahu, picks up his old Japanese pals, and they all return to Japan… where they get thrown into jail for a couple of years because the Japanese shogun think they could be spies. Oops.
They're eventually freed and Manjiro returns to his family. Not long after, though, he gets called back to one of the lords (daimyos), though he's not sure why. Is he going to be thrown back in jail for being a "spy" or will he become a samurai because of his knowledge and experience with Westerners? The epilogue tells us that it's the latter: Manjiro becomes a samurai and even gets Japan to end its isolationist policies; he continues to do great things up until the end of his life.