Getting the shutter speed right
One of the first things to do to prevent blurry images is to make sure you're shooting with a fast enough shutter speed. Most shooters learn the "reciprocal of the focal length" rule, which says to choose a shutter speed that is at least your lens's focal length as a fraction of one second (300 mm telephoto would call for a 1/500th of a second shutter speed). DSLRs make the math a little more complicated because of the "multiplier" or "crop" effect. Since most DSLRs have less than full frame sensors, lenses designed for film cameras effectively produce an image that has been "multiplied" or "cropped" as a result. (The issue of whether the image is multiplied or cropped is the subject of much heated debate amongst photographers. I'm trying to stay out that debate myself.) Either way, because the focal length of the lens is magnified, the potential blur from camera shake is too. If you're using a 300 mm lens on a camera with a 1.5 multiplier effect, you're effectively shooting with a 450 mm telephoto, so setting that 1/500 of a second shutter speed is even more important. The need for an appropriate shutter speed/focal length combination is true whether you're using a big, heavy, pro DSLR or a credit card sized point and shoot camera that's light as a feather.
In fact, getting the shutter speed right is even more important when you're shooting with the point and shoot than it is with the DSLR. That's because of another important consideration in avoiding, namely the need to exercise proper camera holding technique when making an exposure. The steadier the camera is when the image is made, the less problematic blur from camera shake will be. For DSLR use, proper technique includes bracing the camera properly with the arms and hands and adopting the proper foot positions. It's also important to bring the bracing elbow in tight to the body to improve camera steadiness. (See illustrations for examples.) This technique can be used with any camera that has a viewfinder that the photographer opts to use, even point and shoot cameras.
It's a different story when it comes to using an LCD viewer to operate the camera. Shooting with the camera with outstretched arms and trying to trigger the shutter is a recipe for blurry photos since keeping the camera steady during the exposure is very difficult. Pushing the shutter button will usually cause the camera to move during the exposure, so making an effort to hold the camera as steady as possible is extremely important. Photographers often try to brace themselves and their cameras against a solid surface to help improve camera steadiness. Bracing the camera on a tabletop or desk can also keep the camera steadier. It's also important to trigger the shutter properly. Proper technique calls for gently squeezing the shutter rather than a quick sharp jerky movement.