What is the best camera to use for photography in the price range of $300-$600?

something that will take a great sharp picture without the flash.

Depends:
if you want quality: Nikon D40 (DSRL) for less than $500.
if you want a compact and easy to use: buy Nikon or Cannon
cannon SD890IS around $300
Nikon S560 or a bigger model P80 with zoom of 18x

It depends on your needs. Are you a casual photographer, taking occasional pictures? Or are you the true enthusiast, that understands the fundamentals of photography? If you are the latter, you may want to consider a entry-level dslr camera. In comparison with digital compacts, these cameras offer way superior image quality and manual usability. The downside, however, is that it lacks the ability to record video, is extremely heavy especially when you're carrying extra lens, and doesn't offer the 'point and shoot' convenience of a compact. For that price range, I recommend an higher end compact from the likes of Canon, Fujifilm, and Panasonic. Choose one that offers manual control over your iso, shutter speed, and aperture- That way, it offers room to improve on your skills when you want to. The Canon G9 is a favorite among enthusiasts, offering unrivaled jpeg image quality. Look for the upcoming Panasonic LX3 too, noise is much improved I heard, and nothing beats its wide angle (24mm). If your one of those gadget-happy people who likes to take instant nice pictures, Sony T-series may work out to be the best for you. Its touch screen is one of the best in the industry, and it offers smile shutter and face detection. However, image quality is not a plus for this camera. The fuji fd-series are also really formidable under right conditions, especially the latest one. The test shots from the camera revealed no purple fringing, little distortion, good iso performance, among others. A good way to take sharp pictures without a flash is to use a tripod. Although good-quality tripods can succeed the cost of your camera, it is well recommended if you often take long-exposure pictures. A tripod in the 100-150 range should be sufficient for occasional use. Another way is to raise the ISO, so that the shutter could operate at faster speeds, thus reducing blur. Anyways, hoped this helped! =)

For that amount of money you can afford an entry-level SLR with the kit lens. Take a look at the Nikon D40 or the Canon XS/XT or Canon XTi. Both of these brands are pretty much equivalent to each other so just go with whichever is most comfortable. Once you have an SLR the quality of your photos just comes down to your ability as a photographer.