What to Expect
This is the story of an accidental revolutionary, a computer technician that ends up leading the movement to free the Lunar penal colonies from Earth's control and into independence. Written in the mid 60's, the book is very much a product of its time: back when wearing a shirt with Che Guevara's profile wasn't a cutesy retro thing but a political statement, when putting Russian slang in your novel would get you on on government lists, and when the flower-power movement was peaking. This book is as much about the period it was written and how socity changes since, as it is about the future it predicts.
What I liked
I love Heinlein's story-telling style, which is both unassuming and immediately immersive. The plot keeps rolling and the characters are engaging; you'd want to keep on reading.
His use of language is something I probably missed when I was younger: he not only uses Russian terms, but the protagonist uses English with a distinct Russian grammar (I know a smattering of Russian, but that just increases the joy of reading - it shouldn't stand in anyone's way), with Australian slang thrown on top (you know, that other colonial penal colony). At the time he was writing, this was practically subversive.
There is a certain charming naivete in the futuristic predictions. I'm not referring to having missed the impact that communication networks introduced in the age of information - you can't hold it against Heinlein - but despite this being a story of a war for independence there is still a certain belief in the future of the human race. When one looks at how humanity has both changed and stagnated since it was written, one tends to be more cynical.
What to be aware of
As a product of the 60's, there will be statements that will look chauvinistic and racist to modern readers. That is unfair to Heinlein, as he was actually one of the influencers of the flower-power movement, one of the first Sci-Fi authors to break into main-stream media and have his futuristic work depict racial and gender equality in measures far beyond the times.
Also as mentioned, though his use technology, of AI and computers, is very dated. If you'd like to read a modern work that conjectures about solar-system colonisation, I'd suggest Eric Klein's The One (which, coincidentally, opens up with an homage to The Moon is a Harsh Mistress).
Felix was ambivalent about the book. He felt the natural sympathy towards those downtrodden who seek to throw the yoke of oppression, but at the same time he comes from a culture with a strong veneration of ancestors (and their associated institutions) and where slave revolts are a constant danger. Anarchy, benevolently productive or otherwise, just isn't his thing. Still, he enjoyed the story, and the strive for equality and freedom against rapacious autocrats (something he's definitely seen).
This book, which I've read several times in my impressionable youth, certainly explains my own anarchic and cynical worldview. Despite it having aged in the science department, I'd still highly recommend this to modern readers. It's a book to make you think on many levels: on government, on technology, on things that have changed in how we see ourselves, on what hasn't changed despite what we may think of ourselves. This is a prime example of uncompromising speculative fiction, of using the best of what is currently known to speculate on what could be - and perhaps in doing so motivate a change for the better.