Hi Rob,
The basics really come down to making sure you've got a
good focused picture with decent light, and avoiding image sensor noise. One prerequisite is to use a copy stand or tripod to hold the camera rock steady - and even then you may want to use a self-timer or remote release to avoid any shake as you press the shutter button.
With these
types of consumer point and shoot (I have a 2MP Canon A60 - same deal) you're really fighting the tiny image sensor to get a
good picture. The small sensor size means fewer photons hitting each pixel on the sensor and therefore a lower signal-to-noise ratio than you get with the much larger (physical size, not megapixels) sensor of a DSLR. One thing to be aware of is what ISO settings mean on a digital camera - not quite the same as on a film camera. The image sensor on a digital camera inherently has a fixed sensitivity. What you are doing on a digital camera by increasing the ISO level isn't changing the sensitivity but rather amplifying the signal coming from the sensor to achieve a similar effect, but this has the downside of also amplifying the sensor noise. On a DSLR this is acceptable given the better sensor, but on a point and shoot you want to keep to the lowest end of the ISO range. Manually select ISO 50/100 or whatever is the lowest your camera will do.
For coin photography you're not really concerned about depth of
field, so you really want to choose a wide aperture to let in maximum light, but stop it down a
bit from the maximum to get best sharpness. Try something around F4.0 (a lower number would let in more light, but be less
sharp).
For dpi/image size, there are only really two choices that make sense. Either shoot it at the image size you want, or shoot it at maximum size then reduce it later in
software. If you resize it you may want to sharpen a little to compensate for sharpness lost by the resizing.
Distance from subject to lens is going to be a
bit dictated by your camera and lens - you need to be far enough away for the lens to be able to focus (depends on macro capability - 3-4 inches, maybe), and to give yourself some working room and not block the light, but close enough to get the image size you want. For my setup I can get a full frame focused image at around 3-4 inches; it'd be nice to back off and have more room, but then I couldn't fill the frame.
Type of lighting source doesn't make too much difference (set custom white balance to get true colors for whatever light source you are using), although I prefer ambient light by a window. Light positioning/angle becomes important when you are moving away from the basics and trying to get the "best" picture of a particular coin. Since you'll be choosing the F-stop manually, either use aperture-priority mode to let the camera set the exposure time, or do it manually. A brighter picture will generally allow more leeway for post-processing than a darker one, and will allow for better autofocus which is all that I use.
You're really going to need to post-process to get the best result - largely to boost contrast to compensate for the
poor lenses that come with these
types of camera. If you're using Microsoft Windows then for free
software Google's Picassa does a decent job (the auto-contrast option works wonders!), or you can use the GIMP's levels tool (Tools->Color tools->Levels). Tweaking the contrast / levels curve makes the difference between a competently taken but dull/flat picture and one that pops and looks more life-like.
Hope this helps!
Ben