Another problem with digital cameras is that they tend to need more light than a film camera to create a comparable exposure. As a result, slower shutter speeds are often used to get enough light. This can make it very difficult to take a photo without some blurring. The solution is simple: use a tripod. They're not very expensive, and you can mount almost all cameras to one. This step will result in a huge improvement in photo quality. If you have to take a photo with a slow shutter speed and you don't have your tripod handy, try using burst mode. It will take practice to hold the camera steady (hold it against your face and exhale as you press the button), but the burst will give you a better chance at getting one unblurred picture out of the bunch. When all else fails, a sturdy shelf or stack of books can help.
When in doubt, go for a greater depth of field. With digital images, it is always possible to use software to take certain areas of a photo out of focus, but you can never "fix" anything that's out of focus to begin with.
If your camera has a viewfinder in the upper left corner, you may have problems with parallax, especially for close-up shots. This means that the viewfinder is looking at a slightly different area than what the lens is seeing. Cameras with a through-the-lens viewfinder don't have this problem. Using the LCD screen to line up your shot can help, but the screens still don't usually show all of what the lens is seeing, and they can make it difficult to focus properly.