What is a good way to learn the Parallax Propeller chip?
Whenever I start with a new microcontroller I always go back to a few of my standard projects. The things I enjoy playing with most are RGB LED's, 7 Segment Displays and i2c real time clocks like the DS1307+

My first projects almost always revolve around something I am familiar with like these components.  I might start off with just lighting up the RGB leds. Then I'll get more adventurous and work out PWM on the chip and start making lots of different colours.

This gives me some time with the development tools and I can start to learn my way around the chip. This is just as important as what to make or how to learn about the micrcontroller you are using.

Most controllers these days have a reasonable library so it should be relatively easy to get i2c going. So that instantly opens up a world of add on components to play with. The RTC chips are pretty easy to get going and they teach you a few programming techniques like BCD conversion. 

Seven segment displays are also easy and fun. You can play around with single displays or even use the microcontroller to multiplex additional displays. Not hard to get going and the projects are semi useful and fun right off the bat.

So by learning i2c, seven segment and driving leds. You can now build a very simple clock.

Now I realise these aren't propeller specific, but for anyone starting off it might just give you an idea about what to try!