1.) Buy a Tripod - Buying a tripod does not have to be terribly expensive. You don't need the most expensive kind, just one that will stabilize the image. If you are like me, you've never shot long exposure pictures, which a tripod really helps with. Which brings me to...
2.) Shutter speed - When we take standard photos, it happens pretty quick, usually 1/1000th of a second is what the shutter speed is set at. When we photograph stars, they typically are very dim. We need to set the shutter speed for as long as it will go. Usually the longest you can set this is 15 seconds, but many models have a manual shutter speed that lets you control how long your soon-to-be photo is exposed. The longer the better... to a certain point, so be sure to experiment with your particular model of camera!
3.) Aperture - This is a variable sized hole in the camera that controls how much light hits your digital sensor. You want the the largest sized hole, usually f/2.8. The smaller the number, the larger the hole. f/2.8 is a larger hole, thus allowing more light into the camera, than an aperture size of f/5.6. You want the largest aperture available on your camera to take night sky photographs.
4.) ISO - This setting is usualy set at 100, but can have different default camera settings. Generally, the higher the ISO, the brighter the picture is, however, the result with have more "noise" and appear grainy. More expensive cameras have ISOs of 800, 1600, or even 3200. Try the highest ISO setting when you take photographs and reduce the ISO if the image appears too excessively grainy.
5.) White Balance - This is normally set to automatic by default. Set your white balance to incandescent for much better results. Some expensive cameras have custom white balance filters which can end up looking even better than the incandescent preset. Experiment... but remember, the automatic setting on this will make the photo results less than desireable; especially near light pollution.
The example photo up top is courtesy of Chris Picking from Starry Night Photography. His site has tons of photos of celestial bodies. Check it out; it's well worth it.