First: Turn OFF your flash. Try working totally with natural light for awhile.
First image is flash. Second is no flash. Ok, I admit: my poor, sick son looks miserable in both images. But, can't you see how much richer and full of depth the second image is?!? ;-). And there's no annoying, ugly flash shadow on the wall.
Second: Stop using Full Auto mode, and try instead using Aperture Priority mode. This will give you much greater creative license with your images in terms of what is in focus and what is OUT of focus (the parts you intentionally blur). I think Pioneer Woman has a tremendously helpful explanation of how to use aperture. Throwing a background out of focus helps to keep the focus on your desired object/person/etc.
Here's an example of what aperture does for a picture.
The first image is f/1.8. The second image is f/8. See how the background in the first image is blurrier?
First image is f/1.8; second image is f/16.
Third: Along with using aperture priority, manually select your autofocus point (read your camera manual to discover how) and make sure to put that focus point right on an eyeball (preferably the eyeball closest to the lens). It's a rare occasion when you don't want the eyes in focus!
Fourth: Consider your perspective and get creative with it. I'm not creative by nature, so I have to study other's creativity to jumpstart my own. I find lots of inspiration perusing Flickr.com, a fabulous photography forum where people post their favorite pictures. Look around, see what images move you, and try to reproduce them.
So ... I'm kinda slow. I spent about a year just on step one (turn flash OFF), while still using full auto mode. Then I spent about half a year using aperture priority with manual autofocus points. THEN I began shooting full manual. And then I began lusting after a new camera with higher ISO capability. But, that's another story for another day.
If you haven't already, start playing around with aperture and without your flash. You just might be surprised how much you improve your photography.