I have not taken any course on the above. I am currently reading a book on set theory, and another book on logic. The predicate calculus on the logic is somewhat interesting, but in general logic is a bit too dry. The section on quantifiers is overwhelming. Perhaps, it's just because I am swallowing the book too quickly without ruminating. So far, I sort of know what they are all about, but not enough to do me any good.
From what I see, you can learn set theory and logic at the same time. I don't know what axiomatic theory is.
This is an interesting question you are asking. I didn't think an expert like would ask a question like this.