Adding weights to your camera increases its effective inertia and improves stability. One of the simplest ways to add a weight to your camera is to leave it mounted to your tripod. Use your tripod as you normally would for stationary shots. Then when it is time for a moving shot, just pick up the tripod at the center column and use it as a hand held stabilizer. A standard tripod is heavy enough to effectively counterbalance most point-and-shoot cameras. However, you can add additional weight for even greater stability.
For slow moving shots, grip the tripod near the head. Having the center of mass well below your grip helps to keep the camera aligned vertically. For fast moving shots, grip the center column just above the leg section. Holding the tripod closer to the center of mass helps to keep it balanced during rapid movements.