For those of you using a SLR the best advice I received on learning how to use my camera was to shoot on Manual mode.
To start set the ISO on AUTO.
Try using the smallest aperture F2.8 or F4 (this is for a picture that has a specific focus and the background will be blurry). The lens’ aperture is reduced to improve depth of field and sharpness.
Adjust the shutter speed so it is correctly exposed. In a Nikon if you look in the camera (as you are taking the picture) there is a scale that shows you if it is exposed correctly or not. Adjust your shutter speed until it is. This is how I started learning to shoot on Manual Mode.
DON’T USE A FLASH when learning to shoot on manual mode!
Take pictures in good light to learn. Go outside during the day and find a subject in the shade. As you get more comfortable then you can play with lower light, or harsher light and learn how your settings effect you pictures.
For a Point and Shoot
Put your camera on Portrait mode. Take pictures on this mode for awhile and get close to your subject. Start with your subjects outside, in the shade. Note what is in the background. Make sure it is something you want to look at (eg. not half of a car or a trash can). See what you think.