If your camera has an autofocus mode has the camera track the point of focus and adjust to compensate when/if it moves (like Canon's AI Servo mode), then that's the mode I'd choose.
Assuming you want to stick to auto modes, then I'd choose "Portrait" (icon = woman's head) to blur distracting backgrounds, "Sports mode" (icon = person running) if you're subject is moving quickly (ie a frisbee or agility demo), or "Landscape" mode (icon = mountains) to keep the background sharp.
If the dog's looking at you, you have to be careful with flash. You'll often get "green eye" (rather than the "red eye" you get with humans). You can try redeye reduction mode for your flash (if available) to try to get rid of that, but I find that it often doesn't work no matter what I try. An off-camera flash is helpful in that situation, or just don't use flash at all -if you're outside and can shoot with natural light in mind, you don't really need flash. Just make sure the light is hitting the dog in a flattering way; I tend to look for the best light and look for photo ops in that direction. A good place to start with that is to keep the sun at your back, or if it's close to noon, then try in the shade (mid-day sun can be too harsh to produce good photos).
If you have the space on our memory card and the time to review the extra photos it doesn't hurt. I actually find that sometimes the sound of the shutter makes dogs cock their heads in the "what was that" expression, which is cute and often caught on the 2nd or 3rd photo in a series.
- Get low to get on their level if you can.
- Find a whistle or clicking sound that gets their attention. I find that a long whistle that slowly goes from a normal to a high pitch often gets their attention.
- For portrait-like photos, I tend to frame animals tight and/or blur the background so they stand out and I can eliminate any distractions from my image.
- Wide shots that show the event in the background are nice too, but you have to be really aware of what's showing in the background. Make it tell a story, don't let it just be what happens to be there or you'll likely end up unhappy with the results.