How familiar are you with your camera? Have you ever shot in Tv mode?
Turn your mode dial to Tv and look at your shutter speed to make sure it is not set too slow. Unless you can hand hold very steady, and release the shutter button smoothly, "camera shake" might occur. Depending on your situtation, you might try keeping the shutter speed no slower than 1/60th of a second, perhaps even 1/125th.
If you use the tip of your finger instead of the fleshy area (softer shutter release) underneath your fingernail that might jolt the camera a bit more.
Bracing yourself against a wall, setting it on a fencepost, rock, table etc. and perhaps using the remote with 0 second delay-here avoiding pressing that shutter button will help with your shots.
Finally, get that rock steady tripod if you don't already have one-Bogen makes an affordable one in the $100-$150 range. Don't get the one's from Wal Mart, etc.-plastic camera mount heads ain't tough-they wobble. I can't believe I bought one of those things. IF you use the remote with these cheaper tripods-I realize not everyone has $100+ to spend-and it 's not very windy that will help-you aren't touching the camera.
You say you've shot in Av mode-that is aperture mode-the size the lenses' "iris" opens to let light in once the shutter opens. The aperture reading is on the lower right off-center of the LCD screen, the little green arrowhead points to this in AV mode.
If you shoot in manual mode, you can alternate between shutter speed settings by pressing down on the setting wheel to alternate choosing between shutter speed and aperture-when you press it down the arrow will move from your shuttter speed to aperture reading, then turn the wheel as you need. I really like this feature of shooting in manual mode-it is quick and handy. In the setting window on the top left of your camera, notice the shutter speed/aperture change depending on your choice.
One more thing, higher ISO speeds will give you faster shutter speeds but will also give more noise (grain) in your shots-.
Bottom line, a sturdy tripod and using the remote (or selftimer) might give your sharpest, steadiest shooting. Higher shutter speeds-1/160th and faster will help also.
Try the above-experiment. Write back if you have any questions, I hope this helps you, and have fun with your G3.
Kerry Elizabeth Owen