I recently wrote an article on beginner tips on dog photography. In this discussion, I will jot down a few advanced pointers to help you improve your pet photography.
Use the shutter priority mode
Like in bird photography, the depth of field is not always necessary, but getting sharp photos is. So switching to shutter priority mode makes ample sense. You could also switch to full manual mode, but then you have to control both the shutter speed and the aperture, which can be too much when you have a fidgety subject like a puppy. Switching to shutter priority mode ensures that you have to control just one of the shooting parameters. At the same time, switch to Auto ISO. Auto ISO ensures that your ISO is automatically adjusted when the exposure value gets close to your upper threshold. You selected an upper threshold of 1/500 for the shutter speed. If your shutter speed drops below that, the camera will automatically increase the ISO so that the upper limit isn't crossed. Auto ISO works in all the priority modes and the Manual mode.
Now and then, your subject will allow you to shoot some photos of the subject sitting perfectly still as if lost in thoughts. Those are the moments which you've to wait for. This is because, in those moments, you can experiment with your composition while the dog is perfectly still. I get that with my Labrador retriever once in a while, which allows me to change the composition, play with manual focusing, and play with the exposure. In moments like this, you can experiment with the manual focusing option of your camera.
Shoot with natural light
Pets love the outdoors. My Labrador retriever loves playing in the grass and splashing around when it rains. When he is in his element, it gives me the best opportunities to photograph some fantastic natural light portraits. I can also use the fast shutter speed to freeze moments when he is playing and simply being himself. Always try to shoot against the sun. Place your subject with the sun behind them so you can capture beautiful constant lighting without harsh shadows across the face.
Shoot from the ground level
I have always shot kids from their eye level, which gives a perspective of the world around them from their point of view. I recommend the same when you're capturing photos of your dog. Many beginner photographers are guilty of shooting from the eye level, and that's a lazy way to photograph pets (and children). Shooting from the eye level ensures you're capturing more natural photos.