1. Understand settings
Most of the time the auto settings will produce acceptable exposure results. If you want that extra control the next best setting is Aperture Priority. (Aperture is the iris setting which controls the amount of light reaching the sensor, ( or film). Depending on the brightness of the scene, a high aperture number, say F22, will produce sharpness from the foreground to the background. If you want to concentrate on part of the picture..as in portraits, a small aperture say F4 will give a narrow plane of sharpness so that the face is in focus and the background blurred. (Remember that the iris is widest at a low number, say F4.5, when it allows most light through; conversely the iris is narrowest at high F numbers, so high F numbers tend to be used with flash or in bright midday sunlight.) Warning: If high f number selection results in low shutter speeds, rest your camera, or use a tripod to avoid blurring due to camera shake. In general only use Shutter priority if you know you must catch that speeding car or wish to record special effects such as flow blur on a waterfall. There are many other settings, night time, close-ups, overcast sky, sunny sky etc.
Decide what you want. Try all the various settings. Check the results. Know your camera. (Yeah..I know it's easier said than done.)
2. Think about verticals, horizontals and framing
Certain subjects like tall buildings and faces tend to cry out for portrait shaped shots. Rivers and shore scenes usually look better as landscape images, but photographers can always make a choice and the best shot doesn't always result from the first impression. Think about the best shot and take it but don't be afraid to move in or out to frame the subject or find that detail close up of a person... or that leaf or feather on the ground. Whatever it is make sure you have a choice of views; full frame, distant and close-up. You may not have another chance.
Always try to find both landscape and portrait views. If you're not sure which is best, take both and as many as you can. You can choose the best one later.
3. Use the foreground (a rule of three) and check the frame.
Make sure you have something in the foreground, something in the background and something in between for scenic shots. The foreground is usually the most critical. Use texture such as rocks or cobbles or flower patterns or something colourful in the lower third of the picture. Decide whether you want a clean edge to your image or whether you wish to use an opening, the branches of tree or a window to frame the image. Unless you want the 'framing' try to avoid branches, lights or other intrusions on the edges of your picture if you can.