Why Av?
The pictures you get will look absolutely identical down to the very last detail whether you shoot in M (manual), Av (aperture priority), Tv (shutter priority), or P (auto) as long as the aperture and shutter speed the end up being used are the same. The mode doesn't really affect the exposure (how bright or dark the picture is), it just helps pick the aperture and shutter speeds values that you use.
With Av mode, you control the aperture and the camera picks the appropriate shutter speed for that aperture based on the amount of light the meter says is available. With Tv mode, it's the opposite - you pick the shutter speed and the camera picks the aperture. With P, it picks both and with M you pick both.
So the question is really, what do you want to control and what do you want to let the camera control? If you want to control the aperture (and hence depth of field), shoot in Av. If you want to control the shutter speed (and how motion is frozen or not frozen), shoot in Tv. If you don't want to worry about either and want the camera to use it's judgement, use P. I would only use M if you plan to frequently overrule your meter (probably because you are using flash) or because you want to set an exposure and keep it consistent for a series of shots.
I split my time in the parks between Av and P mode. When I'm going to be somewhere for a while, like a show, I use Av because I want aperture control and consistency. I'm often trying to achieve a particular DOF narrow enough to isolate my subject but wide enough to minimize focus errors ruining shots.
I switch to P mode when I'm roving between areas because the conditions and subjects change a lot. I don't want to necessarily worry about my settings for each of those shots. When I do, it's trivial to override the camera's choices with program shift.
I use manual (actually bulb) mode when shooting fireworks. I also use manual in extreme low light situations. In those cases, I set the aperture as wide as possible, the ISO as high as possible, and the shutter speed as slow as I'm willing to dare. That sometimes leads to underexposed shots that look rather noisy when the exposure is boosted, but that's better than a well exposed shot that is blurry because there wasn't enough light for a faster shutter speed.
