If you can, shoot your subject from different angles and with different lighting. If you can manually turn off your flash, do so. Other flash settings can also be tried such as 'fill flash.' Sometimes even a perfectly exposed picture can be improved with fill flash. Other times turning off the flash leaving the subject underexposed can add an element of drama.
Try taking the same photo three different ways, with three different exposure options. The result can be three very different pictures from each other, but all will look good. You never know if you don't try.
If at all possible, bracket your shots, with one shot slightly underexposed, one slightly overexposed, and one "just right." Some digital cameras will do this automatically, but even then you need to practice. If you are expecting the camera to shoot one frame and it shoots three, there is an excellent chance the last two will look like they were taken by a very surprised photographer.
Special Features: Many digital cameras now come packed with some interesting special features. Examples can be found in the ability to take short QuickTime clips and record sound annotations. Some cameras will even let you experiment with special effects like shooting in black and white or sepia toned images. Focus on the basics before you dive into special effects.
Remember that some of these effects such as black and white or sepia toned images can be created without the camera. Photoshop can transform a color image into black and white in a few simple steps. Basically, if you can do it easily in Photoshop, focus on taking a top quality color photo behind the camera and take care of the special effects without it.
On the other hand, you should try out other special effects. For example, if you pan your camera to track a car moving at high speed, the car will be in sharp focus but the background will be blurred, making an interesting picture. Or you can do the opposite: focus on a particular stationary object -- a child flying a kite, a freshly-painted fire hydrant -- and allow a speeding car to enter the frame. You'll then have a sharply focused center of attention with the added benefit of motion.
Silhouettes are another nice special effect. Try taking a photo with your subject in shadow, eclipsing a brightly-exposed object in the background. Now try the reverse, a brightly exposed subject against a dark background. Although different, both can be interesting. Getting a good silhouette with film is expensive: you shoot a lot of frames with little or no reward. With a digital camera, however, the only cost is your time and patience, and your patience will be rewarded.