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For one thing, the Powershot S3 is 6 megapixels. That in itself isn't a problem (my DSLR is 6MP), but coupled with the small sensor size and reduced clarity of the built-in lens and you're not going to be able to get the same kind of sharpness from that camera. At least, not without some software tweaking of the digital image.
Almost all digital files need some sharpening. Not much, just a little bit extra. It never hurts to have a really sharp original, of course.
As you say, you may be able to help by stopping down the lens for a broad depth of field, but this doesn't always lend itself to a nice photo.
Are you using the focal length reciprocal method of judging shutter speed (50mm = 1/50 sec maximum shutter speed for handheld shooting, 100mm = 1/100 sec, etc)? If so, make sure you are using the 35mm equivalent focal length. That camera uses a 6-72mm lens and if you use the reciprocal of 6mm it will be far too long a shutter opening for handheld shooting (the 35mm equivalent of the 6mm wide end is 36mm). If you are shooting with super long shutter openings that may be contributing to motion blur in your images.
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