Does anyone have any recommendations for a new camera (see details)?

For the past three years I have been very interested in taking photography, however, our family camera takes lousy, low quality shots. I've been saving up cash to get a better quality camera, but as I'm new to it all, I'm not quite sure what kind of camera I want. Above all I need a camera with good image quality, noise reduction, and a good focus. I do not want a camera that would be dependent on a tripod. At this time, I am not terribly interested in a camera with interchangeable lenses. As to my price range, I am willing to go to 450, maybe 500 dollars. I understand that its a lot to ask, seeing that so many cameras can be thousands of dollars, but I just need something that is exceptional for its price range. Does anyone have any recommendations?