Think about Light "He spoke with the wisdom that can only come from experience, like a guy who went blind because he looked at a solar eclipse without one of those boxes with a pinhole in it and now goes around the country speaking at high schools about the dangers of looking at a solar eclipse without one of those boxes with a pinhole in it."
-- Joseph Romm
My personal definition of
photography is "the recording of light rays." It is therefore difficult to take a decent picture if you have not chosen the lighting carefully.
Just say no 
Just say "no" to on-camera flash. Your eye needs shadows to make out shapes. When the light is coming from the same position as the lens, there are no shadows to "model" faces. Light from a point source like the on-camera flash falls off as the square of the distance from the source. That means things close to the camera will be washed-out, the subject on which you focussed will be properly exposed, and the background will be nearly black.
We're at a theater. Can't you tell from the background? That's me in the middle. The guy with the flat face and big washed-out white areas of skin. Part of the problem here is that the camera was loaded with ISO 50 film and therefore doesn't capture much ambient light (i.e., the theater background).
Virtually all point and shoot cameras allow you to control the on-camera flash. What you want to do most of the time is press the tiny lightning bolt button until the "no flash" symbol is displayed. The "no flash" symbol is usually a lightning bolt with a circle around it and line through it. Now the camera will never strobe the flash and will leave the shutter open long enough to capture enough ambient light to make an exposure.
A good point and shoot camera will have a longest shutter speed of at least 1 second. You can probably only hold the camera steady for 1/30th of a second. Your subjects may not hold still for a full second either. So you must start looking for ways to keep the camera still and to complete the exposure in less time. You can:
- look for some light. Move your subjects underneath whatever light sources are handy and see how they look with your eyes.
- set a higher ISO sensitivity, e.g., ISO 400 or ISO 800 (currently only Fuji F30 and rather expensive compact digicams are designed to give good quality at higher ISO settings; the rest just give you a lot of digital "noise")
- steady the camera against a tree/rock/chair/whatever as you press the shutter release
- leave the camera on a tree/rock/chair/whatever and use the self-timer so that the jostling of pressing the shutter release isn't reflected on film. This works well for photographing decorated ceilings in Europe. Just leave the camera on the floor, self-timer on, flash off.
- use a little plastic tripod, monopod, or some other purpose-built camera support
