It might sound obvious, but lots of people don't consider it: get rid of all the ugly things in the background of your photo. Sometimes you can move the thing, and sometimes you can move yourself so the thing is out of shot. Small changes in camera angle can often remove enormous undesirable things from the shot.
Another thing you might be able to do is force the camera to change its Depth of Field (DOF, or how thick the slice of the scene is that's in focus) to make the background blurrier and hide the ugly details from the viewer. There are a couple ways to do this.
First, back away from the subject and zoom in! For a given aperture (which is the size of the hole in front of the lens that light passes through determined by the f-number), zooming closer to the subject will narrow the DOF and give you a blurry background.
Secondly, if you can do it, coax your camera to change the f-number to something lower. This will widen the aperture and narrow the DOF. Not all point and shoot cameras can do this, but if yours can, it's buried in the menu somewhere. Maybe you'll have to set the camera to "Av" or "Manual" mode, if that's possible.