The first step is to turn off that automatic flash. Despite the incredible technology behind digital cameras, they're not very smart about when to really use flash. Any time it's even remotely dark, the flash is used and everything except for your friends' faces becomes washed out. When you turn off the flash, the camera will leave the shutter open longer in order to capture any ambient light. But without the flash, you need to be resourceful so that your photos aren't completely dark. Move your subjects to areas where there is some light. Set the ISO to a higher sensitivity to either 400 or 800. (Unless you have a quality camera such as the Sigma DP2x, you will get too much 'noise' from a very high setting.) Camera tripods or using the self timer is a good option too so the shutter release is steady.
When is the flash useful? When the background is much brighter than the foreground. For example, someone standing in front of a sunny window. The flash will fill the exposure so there is light on the subject and the background.
Another feature that needs to be turned off is the digital zoom. Most are fairly terrible. Instead, you want optical zoom. The difference? Optical zoom is when the elements in the lens actually move to bring the subject closer. Digital is not really zoom at all. It's the camera just enlarging a portion of the image and simulating an optical zoom.
Just about every point-and-shoot comes with preset modes. These are actually great to use because the camera will do all the work a professional will do to take the same shot. Use the sports mode when your kid is playing basketball and the fireworks mode on July 4th. They're a resourceful feature.
The best photographs do not have their subject right in the center of the shot. However, that happens to be where the sensor on a point and-shoot is. How do you solve this problem? Point the sensor at your subject, hold the shutter release halfway down, and then move the camera to how you want the composition. Regardless of whether you do this technique right or not, amateur photographers must stop taking pictures with their subjects in the dead center. It makes for very boring photographs.
Just like their DSLR counterparts, point-and-shoots fall into specific price points. The best advice is to think about your budget and how often you use the camera. If you're willing to spend $400, you can find excellent cameras in that range. But if you want higher quality and aren't too concerned with price, try bridge cameras which are compact, yet have much better lenses and are a step below DSLR cameras.