Should we be training all of our employees in the system standard (such as ISO 9001) we have adopted?
No, employees need first to understand their management system. Only the system managers, process analysts and auditors need to understand the system standard(s). The system standard is not a substitute for the system. Avoid confusion and doubt by running an Awareness Leaders Workshop for the leaders (by job title and influence). From this workshop you will have leaders ready to explain how the system works and why it is important. You will also have an orientation package for new employees.