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Nothing wrong with using Auto settings when you first buy a camera - just to ensure that any important pictures you take turn out. As you become more experienced with the camera (or more experienced as a photographer), you'll move to P-Mode (which is a lot like Auto but it enables you to retain many of the settings as you change them). However, if you need to ensure that your shutter speed it fast enough... or if you need to use a wide aperture, the Tv and Av settings may select an undesirable setting. Hence there's a Manual mode. For controlled work on a Tripod, or when working under known conditions... Manual Mode will be a benefit. For most of the time, Tv (Shutter Priority Mode) will be the setting most photographers use so they can control how fast or slow their shutter will be.
With most compact cameras, I keep the camera in P-Mode. More recently I've taken to carrying all my cameras in Tv mode and I set the shutter to the minimum I need before storing the camera.
With a wide, fast lens, you can usually handhold down to about 1/13th of a second if you have steady hands. But to prevent movement from you subject (people walking etc), you'll want at least 1/60th of a second. If you want a perfect freeze of a car or a wave, you'll want 1/250sec or higher. Each lens is different but if you have a 100mm lens, you ought to set your shutter speed to at least 1/100sec to reduce the effects of camera shake. A 200mm lens requires at least 1/200 sec. However, some of the latest lenses have a new generation of Image Stabilizer which are extremely effective. In these cases you can take a 400mm lens but shoot it at speeds as slow as 1/10th of a second if you have to. Of course, those speeds will be useless for moving people.