Get a book about photography, read the manual and mainly - shoot away! I have been shooting DSLR for about 4 years now and I am still learning, got a D7000 and I absolutely love it! Be warned tho, it can turn into a very expensive hobby
Unlike above I would discourage you from using the auto settings. You may have well sticked with a point and shoot if you want the camera to sort everything out for yourself and give an OK result. Aside from the auto there should be 4 main settings: P, S, A, M.
M is fully manual, which I think you can leave for now. It is handy when you need total control over the exposure etc, or when you shoot a series of photographs and you need them to be all the same (like in case of time lapse videos)
P is what I would suggest you start with. It calculates the correct shutter speed and aperture for the given settings and light. It allows you to change ISO (low ISO = good quality, but needs a lot of light, high ISO = picture quality gets worse but allows you to take sharper pictures when it gets a bit darker - like indoors...), you can also adjust your flash and all other settings.
A is my favorite, it is so called Aperture Priority and it lets you choose your preferred aperture, shutter speed is calculated automatically for you. Aperture has a great deal of affect on what you picture will look like. The 18-55 kit lens you probably got with your camera will have minimum aperture f3.5 at the wide end and f5.6 at the zoom end. The lower the f number the more light can pass through the lens per unit time. This means that at f3.5 you can use faster shutter speed than you could at f5.6. So in low light you want your f number to be as low as possible. High f number on the other hand increases you depth of field, so you get more stuff in the picture in focus. Just turn the dial to A, leave the aperture at f3.5, take a picture of some close object and look at the background. Then take the same picture with f16 - the picture may be blurrier because your shutter speed was slower, but you should be able to see the difference in the depth of field - that background will be a lot more in focus now.
Finally S is your Shutter Speed priority setting. It allows you to adjust the shutter speed while the correct aperture is calculated for you. Shutter speed is measured in seconds and fast shutter speeds (eg 1/1000) are good for taking sharp pictures (hand shake will have less of an effect because the shutter is only open for a very brief period of time), and for freezing motion, while slower shutter speeds can be used to generate "motion blur". This is popular in motorsport where you can track the movement of the car, and have the background blurred. It can also be used when you take pictures of running water like I did with the picture below.
There is a lot to learn when it comes to photography, but it is a very rewarding hobby!
