Camilla and I have just done a bicycle maintenance course with Oxford Cycle Workshop Training (a member-owned cooperative). This only covered the basics - tyres and puncture repair, cleaning and balancing bearings, checking and adjusting brakes, and adjusting gears - but gave me a much better feel for the parts that make up a bike. I'm not sure that I'll do my own bike maintenance (other than the very basic stuff), but it's good to know this stuff.
I highly recommend the course.
While bicycles are surprisingly simple - normal bicycles have to be maintainable by non-specialists - they've been refined over a long period of time and seem to have converged to an elegant transport solution. It's relatively easy to design new kinds of bicycles (and there are a few around), but most adhere to a fairly standard design form. This suggests we can have some confidence that that standard form is something close to an optimum. (Whereas with cars, regulatory constraints make new designs much harder - consider the trauma involved with getting rid of a steering wheel.)