A search of this subreddit shows that people have suggested Soviet Space Cultures: Cosmic Enthusiasm in Socialist Societies and Red Rockets' Glare, but the first book is $60 on Amazon and the second isn't available as an ebook (which means I won't read it).
Any other good suggestions?
I can recommend Matthew Brzezinski Red Moon Rising: Sputnik and the Hidden Rivalries that Ignited the Space Age (2008). It's a political history, not a technical one, of the pre-Apollo space race and tells the story from both sides. As an Americanist much of the detail on the Russian side was new to me, and I thought it was presented well.
It's also just $1.95 used on Amazon.
Red Moon Rising: Sputnik and the Hidden Rivalries that Ignited the Space Age (2008) by Matthew Brzezinski is a popular history book covering the opening years of the US-Soviet space race. Definitely not an academic work (Brzezinski is a journalist) and thus tends to read more as a narrative of the bigger personalities involved, almost melodramatically so, but a good summary of events between 1945-1957. Currently less than $10 for Kindle; not sure if it's available from other ebook vendors.
It's been a couple of years, but if I recall some of the main topics included:
The 1945 race by US and Soviet forces to collect German rocket scientists.
The origin of both nations' space programs as military intercontinental ballistic missile programs. In particular the Soviets' focus on missiles as they realized they couldn't compete with the US on intercontinental bombers.
Stalin and Krushchev's influence over the USSR's space program, the major internal Soviet political debates about the space program. Similar politics within the US military and political leadership.
Inter-branch rivalry within the American military. The federal government was highly interested in developing rocketry, and all three branches made justifications for a share of federal funding. The nascent US Air Force argued anything flying through the sky should be their domain; similarly the Navy argued anything flying over oceans (i.e., to strike Soviet targets) was theirs. Not to be left out, the Army argued rocketry and ICBMs were an extension of artillery--their domain.
The main development teams, e.g. Werner Von Braun and his fellow German "prisoners of peace" on the American side, and the Soviets led by the "Chief Designer", Sergei Korolev.
Its coverage ends with the successful launch of Sputnik I, so you'll have to look elsewhere for a deep exploration of the Soviet manned space program.