Good tip though if I ever find myself in the possession of a more sophisticated camera.
(Note: "perfect" means people will ask you how you did it, and be suitably impressed at your photography skizzles)
Firing flash along the axis of the lens is a poor idea almost always. Also, your pop-up flash has an effective range of ~10 feet (depending on camera), so popping it beyond that range (which I see many people do) is basically pointless. In fact, if there are objects near, what the camera will do is expose for those objects, leaving your main subjects, who are too far away, in shadow.
This is why you'll see those crazy pictures where everyone is real dark and has glowing eyes.
But all in all, the solution is to get a faster lens - there are many point and shoots that have premium, fast lenses. Going for those will allow you to skip the flash for a lot more occasions, making better pictures overall.
I recognize that my P&S camera isn't fast enough to take pictures in low light without the flash, but are there other situations where I should be using a flash?
Generally, using a flash makes my pictures look "cheap" (better than blurry, I suppose). Even when there is strong backlighting and I need to use the flash to balance it out, it still looks wrong.
Much to my wife's consternation, I usually tend to push the no-flash as much as possible (I've missed quick-moving children and other shots because of this). I suppose if I used the flash I would have actually captured the moment, but I don't think I've really seen a picture I've taken with the flash that I've found to be remarkable looking.