I've never liked Heinlein's work, and this is one of the worst. I still read it all again when someone else only got partway through. Like many Heinlein books, he brings lectures and makes a framework to hang them on.
There's a Lunar Colony that is mostly convicts sent up from Earth. There's some scientists, too. The authority would be happy if they were dead as long as the grain kept being catapulted to Earth for their masses. Our protagonist, Manny, accidentally burned his left arm off and has replacements that he uses when necessary. He's primarily a computer tech and he's the one who finds out that the computer, Mike, is sentient. He meets a woman named Wyoming, called Wyoh, and a professor, and there they were -- the first cell of the revolution.
We see a rage of lots of people try for revolution and never get it, but the prof helps Manny and Wyoh form cells. Mike is pretty much everywhere on Luna and can not only report back, but calculate things. When the warden and the guards are killed, they have to move up the time for the revolution. Manny and the prof take to the US in a moderately-padded grain bin from the catapult. They tried to get all the nations to recognize Luna as a real separate country.
We find out that one of the reasons that the prof brings Manny is so he can be inept and annoy people. When there's enough annoyance, Stu, who had been in Luna for a short time, joined them on a trip to Luna using a ground-to-orbit ship. The Loonies have to do something to change and then Earth and the Authority starts sending up ships. The Loonies have already figured out how to use materials to bring them down, and they do. They lose some men, but almost all of the Earth soldiers are dead.
Luna tells the Earth that all their people are dead, but Earth says that's not true. Mike starts throwing metal-encased rocks to very specific places on Earth and two more ships come to Luna. One knocks out the catapult, but the Loonies built a secret one and continue bombing with rocks. Both ships were destroyed. After throwing rocks a few times around the Earth, the nations quickly started recognizing Luna as a separate nation and eventually all of Earth gives in.
Mike isn't talking to him, and Manny has to do some things first, but when he reaches Mike, he doesn't seem to be sentient any more. He talks and is still brilliant (although part of him was killed by the Earth ships), but isn't really Mike anymore. Manny is still thinking about it much later.
The Cat Who Walks Through Walls is not a sequel, but Mike is worked on.