Trying to attain limited depth of field in digital cameras with small sensors (like our G3s) is somewhat of a problem. You have the right idea -- using a large aperture (small F stop number like 2, 2.8, etc) but the effect is not as as obvious as in a 35mm film camera. To get the blurred background, you really have to force the largest aperture (smallest F stop number) possible.
Put the G3 in Av mode. Zoom to maximum telephoto setting. Set the smallest F stop possible. Use the LCD panel to confirm the focus point and shutter speed that the G3 will automatically select. Focus on the subject and notice the shutter speed. If such a low shutter speed is selected that the G3 gives the "shaky" warning, then put the camera on a tripod or otherwise brace it and take the photo. Take a second photo but this time focus in front of the subject.
Exposure-wise, for a given ISO setting and amount of light, a fast shutter speed matches a large aperature (small F stop) and slow shutter speed matches a small aperture. For example, if you were to force F 2 and the corresponding shutter speed were 1/30 second then if you forced F4 the corresponding shutter speed would be 1/8 second.
If there is ample available light, the G3 ISO number should be at "50" when using the larger apertures. In fact, it is really best to use the ISO 50 setting whenever possible to maximize image quality.
If there is a lot of available light, even the fastest G3 shutter speed may not be fast enough to force F 2 -- perhaps that is your problem. In that case, be sure to set ISO 50, then go to the G3 menu and turn on the internal "ND Filter". The ND (neutral density) filter is a light absorbing filter that intentionally reduces the amount of light striking the image sensor. It is a "three stop" filter so that if the exposure would have been, say, 1/1000 sec at F2 without the ND filter, the exposure with the ND filter would be 1/125 second at F2.
Hope this helps. Good luck with your experimenting.