Choosing your camera
Decide first whether you want to start small or go big. You might decide to go small because you aren't sure you are going to like a DSLR and don't want to make a huge investment in something you never use. Your budget might push you there. You might want to go big because you have the money for a big purchase and believe you will use it a lot. There is nothing at all wrong with either of these choices.
I went small. I found a small, local used and new camera store in my area (Focal Point Photography in Dallas, OR for those in my area who may be interested). I went into the store knowing my budget. I talked to the owner of the store for a while, letting him know what I planned to use the camera for (indoor high speed sports like flyball being my main interest), what my experience was (nill, in regards to DSLRs), and what my budget was. He hooked me up with a Canon EOS Rebel XT and a 50mm 1.8 prime lens. He also spent some time going over the basics with me and showing me how to use my new camera. That part of the interaction was invaluable, and made it well worth going to a camera shop rather than purchasing a camera off Craigslist or similar.
First and most important (and stated everywhere in basic photography guides) is to take you camera off the Automatic settings. I won't go so far as to tell you to put it on Manual all the time, though. I didn't. When I tried, my pictures were horrible and I got discouraged. Instead, try starting out with Program. It is like the automatic settings, in that it will choose aperture and shutter speed for you, but has more flexibility. Mostly, you can choose your own ISO and your own focus point, which you can not do in the automatic settings. Program mode will help you get some good photos while also helping you learn what you are doing.
As you start off taking your first pictures, pay attention to the settings that the camera chooses. If you really love how a photo turns out, make sure to note the aperture and the shutter speed. Paying attention now, when you shoot in Program, will help you out later, when you are trying to figure out what settings to use in a certain situation. Faster shutter speeds are not always better. Large apertures (small numbers) are not always better.
At some point, you will likely notice something you don't like in Program. Maybe the camera is choosing too slow of a shutter speed and making your fun action shots all blurry. Maybe the aperture is not giving you the nice blurred background you were picturing when you set up a shot. This is the point where you will want to start experimenting with your manual modes. There are three of them: Shutter Priority, Aperture Priority, and Manual. Shutter priority has you setting the shutter speed, but the camera choosing aperture. Aperture priority is the opposite, with you choosing aperture and the camera automatically deciding the shutter speed. Manual is where you are choosing everything, including aperture, shutter, and ISO.
If you enjoy a structured learning situation and are looking to move out of Program, this 31 Day to a Better Photo course is awesome. It is free, a fun challenge, and a good way to learn a ton about your camera. I wouldn't personally do it until you get more of a feel for your camera and are comfortable at least messing around with the different settings, but just want to improve your overall knowledge and quality of photos. You could jump right in the day you pick up your camera, though, if you are willing to spend time reading your manual and really work on it. If you are a blogger, you can even post the different challenges and see if your readers want to follow along with your, or start a blog hop with other bloggers.
You don't have to do a specific learning course, though. You can teach yourself through trial and error. Just make sure that you don't miss a once-in-lifetime event (like a wedding) by sitting there messing with your camera buttons!
Say you are taking pictures of your dog playing fetch or running/playing. Your camera is choosing a shutter speed that is leaving the dogs blurry. So, you take the big plunge and switch over to Shutter Priority mode for the first time. Before you switch over, make sure to look at what shutter speed the camera was using in Program mode. Now, flip your dial over to shutter priority and make the shutter faster than it was in Program. You don't have to go overboard. If the camera was using 300 and the pictures were just a bit blurry, step it up to 450. There is no need to go straight to 2000! Of course, you can also just start up at a high shutter speed and then adjust it back down until the lighting is bright enough and the motion is stopped.
You can do the same thing, of course, with aperture. Use the camera's choices in Program mode as a starting off point, and then adjust from there.
At some point, you will probably be in Aperture Priority and realize that the camera is making the shutter speed way too slow, or the other way around. And that, my friends, is when you will switch into Manual.
Patience is a virtue. No one gets amazing pictures right off the bat. Expect to get some totally blown out, over exposed pictures when you forget to put your ISO back down when shooting in full sun at the park after taking a cute picture the night before of your dog settling down for bed. You will miss some shots. It happens. But as you learn and get better, you won't only get a so-so shot of an amazing moment, you will learn how to get amazing shots of amazing moments, and even how to make so-so moments look amazing in pictures.
I wouldn't have believed it when I first got my Canon XT, but I now shoot almost exclusively in Manual. But it has taken me almost a full year and a half to get there. Many pro photographers still use the Priority modes as preference in their shooting. And I still use Program sometimes, if I just need to get the once-in-a-lifetime shot.