What a great idea for a discussion group about photography! I've worked as a portrait/wedding photographer for over 10 years, I thought I'd say a little something about it. I have to tell you, aside from proper exposure, it's all about creativity, and "location, location, location." There is a lot to good photography, but I think with a few tips, it can be fairly simple.
Camera: When I shoot professionaly, I have huge cases of expensive equipment that I haul around, but for most of the shots of my kids and dog, I use a small Canon or Nikon point & shoot pocket-sized camera. The point is that the camera is only a tool, the quality of the picture greatly depends on the photographer as well. In a manner of speaking, the hammer is only as good as the carpenter who swings it.
TIP #1: I Have Seen The Light.... No, I'm not breaking into gospell, but without light, there is no picture. Make sure you have enough of it. Modern digital cameras perform really well in low light, and get better from one year's model to the next, but there's no substitute for real light so make sure you have enough of it. If you're inside try to raise the overall light level or the camera's ISO in low light situations and if necessary lower it it in really bright situations. Flash can be used as well, but it can sometimes be a hard, sharp, or contrasty looking light source because it's such an itty-bitty light source. Just look at how small that little bulb in the camera's flash is in comparison to a 50lb doodle! The results with this tip can greatly vary from camera to camera, so keep that in mind.
TIP #2: Back up and use the telephoto-zoom. Camera lenses can distort facial features when used really close-up. It also may cause exposure problems having the camera and flash too close to the subject and they'll be washed out by hard, contrasty light. You will get a much better perspective by standing further away and then zooming in with the telephoto, then if you just walk right up to the dog a snap a shot right in his face. He'll probably change his behavior when you approach him, resulting in missing the shot. So be stealthy.... back up, zoom in, frame it and shoot.
TIP #3:Keep it Simple. Decide what you want a picture of. It sounds silly but be specific. If you want a close-up of your dog, then make sure that other distractions are not included in the picture, such as your kids' slippers, TV remote, whatever. The simpler the background is, the easier people looking at the picture can identify with the subject, in this case your dog.
Now obviously, if what you want is the dog in some crazy environment or mischevious behavior, than capture that, but just be mindfull of a clean, simple background at all times.