Blogs etc.
"I don't have time for more math classes but like math. What should I read?" --- every student ever.
Well, at least one student asked the question, so here's a long response. Let me know if I missed anything interesting.
Clubs
UM Undergraduate Math Club. Weekly math talks (at the undergraduate level of difficulty) and free pizza.
Michigan Math Circle. Weekly meetings for middle and high school students.
Blogs
Roots of Unity by Evelyn Lamb for Scientific American. Knots, numbers, and math history.
Elements of Math by Steven Strogatz for the New York Times. High-school math from professor's perspective.
What if? by Randall Munroe. Weekly bizzare scenarios by the author of xkcd. Some relation to math.
Math Blogging, a collection of blogs at various levels.
Terry Tao's blog by Fields medalist Terry Tao. The blog that professional mathematicians have trouble reading.
Books
My Best Mathematical and Logic Puzzles by Martin Gardner, whose legendary blog column ran in Scientific American for 24 years.
Flatland by A. Square. Classic book mixing satire and exploration of what it would be like to live in 2 dimensions.
A Mathematician Reads the Newspaper by John Paulos. Noticing math mistakes in the news.
Michael Starbird: various books on mathematical thinking.
Harry Potter and the methods of rationality by Eliezer Yudkowsky. A bizzare and entertaining take on Harry Potter, with logic and citations.
Indra's pearls by David Mumford, Caroline Series, and David Wright. Beautiful illustrations to the topic of Kleinian groups. Starts off easy but gets complicated.
Mathematicians: An Outer View of the Inner World by Mariana Cook. Photos of famous mathematicians with brief descriptions of them or their work.
History of Pi by Petr Beckmann. History and Pi. With some digressions and proofs.
Godel, Escher, Bach by Douglas Hoffstadter. Extensive discussion of various topics including ideas at the core of modern math and logic. Dense book that few have finished, but everyone recommends reading.
Fermat's Enigma by Simon Singh. A history of a famous number theory problem.
The Code Book by Simon Sigh. A history of codes throughout history.
Videos
The pleasure of finding things out by Richard Feynman. Great person and physicist.
Numberphile video-blog. Lots of interesting math for the public.
MIT video courses. Various courses, from calculus through quantum field theory and beyond.
Khan Academy. Basic course lectures.
MSRI lectures. Research-level mathematics lectures from MSRI (Mathematical Sciences Research Institute).
Vi Hart. Whimsical discussions of lots of math, with colorful pictures, pies, and music.
Encouragement
Words of encouragement by Mark Sapir.