How do I start shooting and processing RAW files on my computer?

To get started, set up a few shots from a tripod and shoot them in both JPEG and RAW. Most DSLRs can do that simultaneously but I suspect your camera may not have that option because its writing pipeline is slow (it would otherwise lockup your camera for 20-45s IIRC).

Then load the RAW into any conversion software and see if you can produce an image which YOU prefer to the in-camera JPEG. Play with the conversion controls: sharpness, saturation, contrast, curve, etc. Don't go with the default conversion unless you want to waste your time because that will almost always produce the same JPEG as the camera (some advanced programs will let you define your own conversion though which is usually called a preset).

REMEBER: The RAW advantage is about what YOU can do with the image. Most mediums cannot even show all the nuances in a JPEG (nearly no LCD monitor can), so it is more about having control on the final image than about showing one with more color tones.

After a few rounds, you'll be able to judge if it is for you or not. There will be a cost in terms of space, speed and workflow. Particularly since you do not have a DSLR, every time you shoot a RAW image, it will be a while before you can shoot again. Then you have to realize that if you don't take the time to make the output better than what the camera does, you're not getting much out of RAW. If you do, realize that you could have been shooting more instead. Ask yourself what you prefer and what is worth it.