I'm especially interested in the spaceflight of Vladimir Komarov, the recruitment and preparation of cosmonauts, the USSR's attempts at a crewed moon landing as well as the more technical and psychological aspects of the job of US Apollo astronauts. I know this is a wide array of topics, but I'll be grateful for any help pointing me in the right direction.
In general, these are the best books I would recommend for a history of the Soviet space program:
Soviet rocket designer Boris Chertok wrote a four-part memoir titled "Rocket and People." You can acquire hard copies various places, but NASA has posted the entire thing (in English) as free pdfs here.
Authors Francis French and Colin Burgess wrote a two-part history of the early years of the space race leading up to Apollo, Into That Silent Sea and In The Shadow of the Moon. These aren't focused purely on the Soviet program, but rather are an international history of those years, interlacing the US and USSR missions. The two books are part of the University of Nebraska Press' Ordway-award-winning Outward Odyssey spaceflight history series. (Full disclosure: I'm co-author of a couple of OO books.)
Anatoly Zak is an award-winning chronicler of Russian spaceflight, and wrote a book titled Russia in Space about eight years ago. It may be out of scope for what you're looking for; it skews a little later than the moon-race era. It's hard to find currently, but is slated for a reissue as part of the Outward Odyssey series.