Great words, Sarah. To your list, I would add one more: "protocol".
In the context of "you need to follow the ...",
"process" and "procedure"
are very nearly identical. The difference is that "procedure" has a methodical, almost mechanical, feel to it. Thus, for example, a surgeon follows a rigorous procedure when he operates on your heart. "Process" is a broader word that includes many more situations. A procedure is always a process, but a process is not always a procedure. A process can be chaotic and make little sense. For example, raising a child is a process, a very long one, but it is most definitely not a procedure, unless the child is raised by a computer.
"Progress" is different from the others. It refers to moving forwards. If your child is in school, you follow his progress by watching how he is learning and improving as time goes by.