An issue that comes up often when I talk with people is the desire to buy a better camera to get better pictures. While this is rarely possible, I want to write a note regarding just what you can expect after unpacking that new DSLR you just bought.
First, a DSLR is a hideously complex device. Manufacturers have managed to hide much of the complexity behind auto modes, but you must understand that it didn't just go away. The camera still has the same limited dynamic range, depth of field and sensitivity - as well as other factors that must be balanced in order to get a good shot. All the auto mode does is put the trade-offs in the hands of the camera. Instead of you, the photographer, choosing shutter speed, aperture, sensitivity, focus distance and so on, the camera will use various algorithms to do so for you. Most of the time, the algorithms work; but some times they don't quite result in what you thought you would get.
Just as with any team effort, it takes a little bit of time to get the team members working together. When I bought my Nikon D40[a], it took 103 shots before I got it figured out well enough to be able to reliably get the exposure I wanted. My best advice is to experiment with the metering modes and exposure compensation setting. I ended up setting the D40 to center weighted metering, -0.7EV exposure compensation, and I may have used other settings less than ten times after 40,000 shots. It's all about knowing how the camera "thinks". Having a consistent way of metering is better than having a smart one. I never got the matrix metering of the D40 to work for me. It would mostly get it right, but would get it wrong - somehow - in about a quarter of the cases. Due to the inconsistency, I would never know what the result would be. Center weighted always overexposes by 0.7EV. So, I dial in -0.7EV and I'm set for perfect exposures.
Besides exposure, sharpness seems to be the big deal. With modern cameras we have fairly small sensors with lots of pixels. Take the Nikon D7000[b], for example. It has 16.2 megapixels, on a DX (1.5 crop factor) size sensor. That's a lot of pixels in a small place. To be exact, each pixel is 4.78µm across. With a kit lens (or with anything besides super-pro lens), there is simply no way to focus that well across the full frame; even getting the center of the image pixel-sharp is an amazing accomplishment. If you then watch the image at 100% on your monitor, you're liable to notice that there is some softness. Don't worry about this. All lenses have some distortion and aberration. If you have a camera like the D7000, you must hold the camera perfectly still, have it focused at the perfect distance, have a subject that is perfectly still - and preferably shoot in a vacuum with a mathematically ideal lens. Most ultra-sharp photos have been shot in a studio with very controlled conditions, using lenses that will cost you an average life salary. Alternatively, they have been massively digitally processed to extract sharpness beyond what the photographic apparatus is capable of. Don't expect to get the same sharpness from your kit lens.
What can you expect then? Well, just like with exposure, there's a lot to the focusing. Each lens has an optimal aperture where the distortion and aberration of the lens is at their lowest before their contributions to softness are overtaken by diffraction. For example, a lens can be limited by distortion and aberration from f/3.5 to f/8. At f/8 it is optimally sharp across the whole field of view. From f/8 and onward the diffraction inside the lens dominates and makes it softer. It is also the case that you have a depth of field - a nearest and a farthest distance, between which things will be in focus. Outside of these distances you will have areas that are our of focus. Depth of field increases (the near distance comes nearer and the far distance disappears in the distance) with smaller apertures.
The camera's autofocus system must juggle all of this. First, it must select what to focus on. Second, it must pick the right focus distance in order to overlap the subject to as large a degree as possible. Of course, since the exposure meter is adjusting the aperture it must consider that as well.
Sometimes it gets it wrong, and you end up with a blurry picture. Just as with exposure, I value consistency over smarts. The all-area focusing of the D40 never really made sense to me. Like the matrix metering it would get things right most of the time, but not enough times to keep me from worrying. I set it to dynamic-area AF-A, and have left it there ever since. Coming from a compact camera where contrast-detection autofocus made it possible to focus on anything in the frame, the concept of only being able to focus on specific points was very weird to me. I don't remember when it started feeling natural, but I suspect it took a couple of hundred shots.
There are a lot of other things that can trip you up - flash, flash sync, image stabilization, and so on. It is quite normal to have spend time to learn to use a new camera. It always feels awkward at first.
Cameras also have in-camera picture processing options: Sharpness, brightness, saturation, and so on. I find it easier to just leave these set to default values and do any processing in Photoshop later. I don't have time to play with them when I'm shooting, and as long as I get consistent result I know I'll always be able to correct it.