Jon, I started out with P mode because it allows me to fiddle with a lot of "extra" settings such as flash, white balance (one of the reasons I got a G3) and manual focus, yet still let the camera determine exposure. Later I moved on to Av and Tv, and then manual. I don't use the portrait and landscape modes much.
One trick I use for action photography is to set the camera for Av, and put it wide open. That gives you the fastest shutter speed possible. It also has the advantage that in Av mode, the camera can always find an exposure time that works. That's not true in Tv; if you set a shutter speed of 1/1000 on a heavily overcast day, you'll probably wind up with underexposed shots.
Blurry shots can come from two causes. If everything is blurry, you're getting camera shake. The camera will warn you with a blinking yellow light if the shutter speed is low enough that camera shake is a problem. If you can't use a tripod, try going to a higher ISO. On the other hand, if your subject is blurry but something else is in focus, then the camera is focusing on the wrong thing. Make sure your subject is in the little center box, press the button halfway, and then recompose the frame while holding the button down.