I collected Kim Greene's answers into one, which unfortunately makes it look like the answer came from me. (They are fine answers, just not mine.) My own answer is that in Kim's list, only the Notices and the Intelligencer provide a good graduate/professional perspective on mathematics. However, those two aside, the "magazines" that you read should also include the four best Internet resources for serious mathematics: The arXiv, MathSciNet, Wikipedia, and now this site. There are other fine math web sites out there, but as far as I know, nothing else is as important as these four.