The first decision is whether you just want some family and vacation shots (a point-and-shoot camera will do best and will cost less), or whether you want to try for more artistic photos and really dive into photography. This section is for the latter group.
Whether you use a digital camera or a film camera, there are certain basics which remain the same. Note, however, that with today's digital cameras, for most people digital is the way to go. Our camera club has gone from 98% film in 2000 to 98% digital in 2015.
Compostion: To create good pictures you need good composition. Composition is best achieved by proper aiming and adjustment of your camera settings when you take the photo, combined with an understanding and artistic sense of what makes good composition. The more you do in your camera when you take the shot, the better. Cropping and cloning can be used to improve composition after the fact, but in the process you lose some of the resolution.
Know Your Camera: You need to fully understand your camera. Read the manual, then re-read it, and re-read it again a few weeks later.
The basic settings on a camera include focus, exposure time, aperture setting, and ISO speed.
Focus: Whether you are focussing on your subject manually or using auto-focus, if you do not focus carefully on your subject the resulting photograph will be blurred where you most want it to be sharp. Focus is affected not only by whether or not the lens is properly adjusted, but also by camera movement. If the camera moves while the image image is being exposed, the image will be blurred. Image stabilizing lenses help a lot in this regard, compensating for reasonably small camera movements, and producing sharper images (taken hand-held). A tripod is recommended whenever practical.
Exposure Time: You need a fast exposure time to freeze a fast-moving object, but a longer esposure time can be used to good effect, such as softening the flow of water. Brighter lighting on your subject allows faster exposures, low lighting levels require longer exposures. A wide-open lens allows faster exposures (but has disadvantages mentioned below), while a small lens opening requires a longer exposure time. Photography is a matter of trade-offs.
Aperture Setting (aka f-stop): The aperture is a measure of how wide open a lens is, and is also called an f-stop. As mentioned above, a larger lens opening (which unfortunately confuses new photographers as it is a smaller f-stop number) allows for faster exposure times, and vice-versa. But there is another tradeoff for aperture setting. A wide open lens has a small depth of field in focus, while a small opening in the lens has a much greater depth of field in focus. When working with large aperture openings (small f-stop numbers), the foreground and background will tend to be blurred (assuming you have properly focussed on your subject). This might be helpful when photographing a bird which will look great against a soft green background. Conversely, when working with small aperture openings (larger f-stop numbers), much more of the foreground and background will be in focus. This would be perfect for a scenic view.
ISO Speed: The higher the ISO used, the more options you usually have in terms of trade-offs between aperture setting, depth of focus, and exposure time. Unfortunately, higher ISO images tend to be more grainy, effectively having less resolution. Some of the newer digital cameras (such as the Canon 5D, 7D, Rebel, etc.) have excellent performance at ISO speeds of 400 or 800 that rivals ISO 100 film. Note that we mention Canon in this section: that is because we use Canon cameras and are familiar with them. Nikon and several others also make excellent digital cameras.
Clearly a photographer is juggling trade-offs between film speed, exposure time, aperture setting, and ISO. Read some good introductory books on photography. Practice. It will become second nature with time.
Most of all, have a good time taking photographs.