TL;DR: Put it on the camera and start playing with it.
If you got yourself an OEM hotshoe flash, such as a Nikon SB-600/700 or a Canon 430EX/430EXII as your starter, this is going to be the best place to begin learning what you can do with flash. A lot of folks will urge you to go straight for off-camera and Strobist setups, and that's a great way to "get the most" out of your flash, but if you're a ground-zero level newb to adding light to a scene, the simplest, easiest way to get started is learning how the light behaves with a flash on the camera, and learning to bounce.
The camera mode matters
The first thing to become accustomed to with an external speedlight mounted on a dSLR body is that the shooting mode you've got the camera in causes the flash to behave differently than you may be used to with the integrated flash on a P&S camera in lower light. In full manual mode, you can do whatever you want. In aperture or shutter priority modes, the assumption is that you want to use the flash for fill, so the camera will set your exposure settings relatively close to what you'd have had without the flash, and then throw in enough light from the flash to "fill in" the shadows. In low light, however, aperture priority may lead to very slow shutter speeds, which is not what you're expecting. In Programmable auto, the camera assumes fill in good light, but in low light switches to using the flash as the key, or main light, and acts much closer to what you're expecting. Custom settings on camera bodies can alter this behavior in aperture priority mode, so read your manual.
Your metering changes
Metering is also going to be different than what you're used to with ambient-only (i.e., all the light in the scene that's not from the flash). Be aware that your meter can only measure the light that's present in the scene. Your flash burst isn't there yet--so you have to visualize what adding flash will do to the exposure. It's not just about getting the needle to 0 any more.
I'd highly recommend being comfortable shooting in M mode before attempting to learn flash. Because ambient light is controlled by iso, aperture, and shutter speed. But flash exposure is controlled by iso, aperture, flash power, and flash-to-subject distance. And you're going to want to balance your flash against the ambient. It's easier to learn to juggle five balls while riding a unicycle if you already know how to juggle three while standing still.
Balancing flash against ambient
Balancing the flash against the ambient means that you can use the ambient light at one level, and then change the lighting level on something else using your flash. You can choose to balance this however you want, all the way from your subject as a black silhouette with a white background, all the way to a brightly-light subject with a black background. It's all about the relative levels of light (ratios).
The most typical technique here is to put your subject in the shade, expose for the background (which your camera is likely to tell you will be underexposing by a stop or two), and then add light to your subject to taste. Using a smaller aperture and a lower iso becomes possible with this technique, so your portraits are liable to come out sharper and have more "pop", while your background will be more colorful, rather than washed out.
Bouncing is your other big weapon with on-camera flash. Instead of pointing the head of the flash directly at your subject, and getting flat, on-axis light (i.e., that washed out P&S flash look), you can point the head at a reflective surface (wall, window, door, ceiling, a reflector, someone's shirt front...), and have the light that's reflected off that surface be the main source of illumination. This spreads and softens the light, as well as giving shadows that can bring depth and detail to your subject that flat on-axis light will not. The main thing to keep in mind here is that angle of reflectance = angle of incidence. Think of it like bank shots in pool. Or, more simply, just point your flash in the direction of where you'd put a softbox in the studio. Also be aware that the color of the surface you bounce off may add a color cast to your light, and that your flash can only throw light so far (see inverse square law). Also, while you are pointing the head of your flash at the bounce surface, some light will be leaking directly from the flash head to your subject. You may want to consider flagging that off.
A great starting place to learn on-camera flash is Neil van Niekerk's Tangents website.
White Balancing and Gels
The color of the light from your flash is probably going to be predominantly blue or cooler than a lot of ambient lighting may be. Flourescents can be greenish. Tungsten will be orange. When you throw light from a flash into the scene, white balancing can becoming trickier, because you now have mixed colors of lighting in a single shot. If you balance for the flash-lit bits, everything else in the frame is going to go more orange. If you balance for tungsten ambient, whatever your flash lit is going to get a lot bluer. Sometimes you can use this for artistic effect. Sometimes you don't want to. You can use gels--pieces of transparent colored plastic--over the head of your flash, to get the light from the flash to match the ambient color, so white balancing once again becomes simple. Or you can also use gels to boost the opposite color if, say, you want the background to have an overall blue or orange cast relative to your subject. And when you go off-camera, they'll also come in handy to color your rim/hair lights or backgrounds.
TTL vs. M
You also want to be aware of the mode of your flash. TTL is an automated way for the camera to set the flash power, based on metering. TTL works by having the camera tell the flash to send out a small "preburst" flash of a known brightness level. The camera body's AE system meters this preflash, and then adjusts the flash's power output to what it thinks will be a good illumination level. Just as you use aperture priority on the camera for speed and convenience, you use TTL on the flash. And just as you use M on the camera for consistency and precision, you'd use M on the flash to do the same thing. The most common uses for TTL are event shooting, where readiness and changing lighting conditions may mean more than precision, while M is typically a better tool for studio work where you have time to zero on the perfect settings.
If you use TTL, however, one of the drawbacks for learning is that you have no way of finding out what the actual flash power setting used was--unlike aperture, iso, or shutter speed, your flash power is not recorded in the EXIF of the shot. So, a lot of us would advocate that you put your camera in M and your flash in M, and start adjusting settings to see what the effect of changing any one setting can do for the image.
Once you've hit the limits of on-camera flash and bouncing, then it's time to consider going off-camera and using your flash in a studio-style setup.
For off-camera lighting, David Hobby's Strobist website is the primary resource most of us go to. If you learn better from videos than reading, Hobby has a series of videos on Lynda.com, as well as a DVD of his "Lighting in Layers" series. And there's Zack Arias's OneLight 2.0 downloads.
I'm not even going to try and encapsulate what's involved with off-camera shooting, but the analogy I make is that on-camera lighting is the blue pill; off-camera is the red one, and there's no telling how deep that rabbit hole goes. :)