I'm envious of the physicists for having a website like the APS website. Unfortunately in math, I think expository articles often lag far behind the original publications.
Regardless, here are some sources I use.
Sign up for the ArXiv daily mailings, being specific about the fields of your interest. Always read the abstracts (they're displayed right in the e-mail), and read the introductions of the papers that interest you. This is tough going because many papers are hard to read, but this is a pretty good way to stay up-to-date.
Check out the ICM talks (at least their titles). Though these are not a complete overview of modern math, and though they only occur every four years, they'll give you enough math to think about for more than four years.
If you know the field or topic of your interest, try contacting a researcher directly. I do that when I don't know about a certain field of math, and I've found we mathematicians are quite helpful to each other. If you write an e-mail explaining your background, I bet you will find a good number of mathematicians who are at least willing to e-mail or Skype with you. (Just don't begin your e-mail with "I have squared the circle and proved the rationality of pi.")
Check out videos at MSRI's website (for math in general) and the Simons Center website (for more math-physics stuff). These videos are a great resource, and workshop videos especially contain introductory overviews of contemporary topics.