Heinlein is clear that his Luna isn't necessarily an ideal society (clearer than in some of his other books) but also clear that he wants us to think about the practical consequences of not having much in the way of government, and also to think beyond the Cold War divide which dominated political discourse at the time he was writing. His occasional use of Russian vocabulary is a) generally idiomatic, though one might quibble about the pronunciation and b) a challenge to readers wallowing in the comfortable dichotomies of their own day.
I loved the bits where the central characters plot the revolution - fairly standard revolutionary practice, of course, but interesting to see it so positively portrayed in a mainstream sf novel. My favourite parts, however, are on Earth, the manipulation of the F.N., Heinlein's successor to the U.N., by the Lunar delegation which includes the narrator. In the 1960s, after the U.N. intervention in Congo, it was possible to imagine that a future world government would dispose of coercive military power. It seems pretty improbable now, and indeed Heinlein points out some of the reasons why it is improbable. It's a nice touch to have a small African country be the only one to recognise the Luna government at first, rather as only Nicaragua and Russia now recognise Abkhazia and South Ossetia.
The political discussion is supplemented by one of the great artificial intelligence characters of science fiction. Mike / Adam Selene could have been yet another irritating Heinlein know-it-all character, but here that role is split between him and the intellectual Professor Bernardo (himself rather modest as Heinlein intellectuals go) and the book is much the better for it. So we have a self-aware computer trying to get to grips with the weirdness of humanity, and by reflection showing us ourselves, yet at the same time in innocence providing the strategic leadership of the revolution. It stands the tired old Asimov laws of robitics on their head.
Heinlein's ideas of sex on the moon are, alas, rather less convincing. Sure, gender issues will be rather different with institutionalised polyamory, but he rather over-eggs the benefits of his preferred arrangement, which in his account has no apparent drawbacks (or homosexuals) at all. Stranger in a Strange Land was less pleasant but more realistic on this subject.
The Moon Is A Harsh Mistress won the Hugo for Best Novel in 1967, the year I was born, beating two exceptionally good books which I have read - Babel-17 by Samuel R. Delany and the full-length novel version of Flowers for Algernon by Daniel Keyes - and three others which I haven't heard of - Too Many Magicians by Randall Garrett, The Witches of Karres by James H. Schmitz and Day of the Minotaur by Thomas Burnett Swann. When I reread Stranger in a Strange Land I said I thought it was the best of Heinlein's Hugo-winning novels; I have changed my mind.