Aside from getting your feet wet with a bouncing ball, a typical thing to start rigging/animating would be a flour sack or something like that. To get an idea of what kind of stuff can be done with this, go rent Disney's "Aladdin" and observe the way they gave humanistic qualities to the flying carpet. Or look at any commercial where a product is animated. It's a good way to get started...the rigs aren't too complex...and it's a good real-world application.

Aside from that, if you don't want to rig and want to concentrate on the animation itself, search around for a simple rigged character (as in made from primitives) and have at it. If you visit the animationmentor site, you'll find links to student blogs that show off what they do with simple characters which may serve as some inspiration.