How would one begin setting up verbose (headers, body, status_code, etc) logging of all inbound and outbound HTTP traffic to K8s services?
Well, body is kind of excessive, but I think that for a state of the art solution, what you're looking for is tracing. Take a peek at OpenTracing and Jaeger which are solid CNCF backed choices. You can use things like Istio or Linkerd to help feeding tracing solution like Jaeger as well.