From the Earth to the Moon is a brilliant TV series that shows it all really well.

I love the episode where they sit down and list out the ~10 things they'll have to figure out how to do in order to achieve Kennedy's promise of landing within the decade.

Then they just assign teams and get on it, working on each item until they can actually do it.

I really enjoyed Sunburst and Luminary by Eyles, from the perspective of one of the young students who stumbled into the space programme and ended up writing a lot of the code for the LM.

I also found Go, Flight! enjoyable. It is about the recruitment and culture of the young flight controllers ("ground control") who figured out as they went how to direct and orchestrate missions in space.

Both are from a specific set of people's perspectives rather than an attempt to summarise. With some care on what one accepts as truth I think that's the more intriguing way to follow the program.