As a n00b, what's the minimum I need to have to play with an Arduino? Will generally any Linux / Windows / Mac computer be able to run the software and connecting hardware I need to program it?
Now if you want it to do something you can see other than in the logs from within the IDE, that's when you start needing more stuff. Generally a breadboard, wires, and some components like leds and resistors come next; for making blinking lights, who doesn't like those?
As far as the computer, it can be best answered here but is just about anything you have: http://arduino.cc/en/Guide/HomePage in the getting started section.
Also, check out for local hackerspacers, as there will definitely be people that to set you on the right path.