Where is a good place to start?
Probably a better idea would be to get started with something like an Arduino. Not too long ago I read about Tessel on HN which seemed interesting, sort of an Arduino but you could program it with JavaScript, if that's your thing.
Also there's tons of articles and videos on the web of fun projects people built with Arduino's and Mindstorms, so I would definitely google around a bit to see if you find something you think is interesting. Anyway, good luck.
An alternative or complementary approach is to work in simulation, and with that decision you can immediately start working with state of the art stuff, as follows:
ROS is the dominant Open Source robot platform. It works with the powerful Gazebo simulator. It's all free. Researchers, companies, universities, and hobbyists all use it, and there's an active community. The learning curve can be steep, but this is powerful stuff.
For a simpler start, take a look at the platform that was popular before ROS overtook it: Player/Stage
robots.net
http://www.societyofrobots.com/robotforum/
shoot me an email if you want to talk robots.