Quickly summarizing the table of contents and some of the internal content, these seem to be the main takeaways:
• Create design principles that stress simplicity.
• Test exhaustively.
• Develop mission rules, procedures, and plans for potential issues ahead of time; iterate as new information comes in.
• Execute rigorous post-mortems.
• Prioritize flexible yet disciplined management.
It's notable that this was published before Apollo 13. It'd be interesting to see what changes Kranz et al. would make to these articles after that.