Not really a spell, but I've recently developed the skill...
First connect with your dog. You can do this by visualizing a blue string of light linking the top of your head to the top of your dog's head (Christopher Penczak's technique), or vividly imagining that you are a dog yourself (Sonya Fitzpatrick's technique), but I do it by feeling out for my pet's aura.
Next, listen... pay attention to visions popping up in your head that's "off" of your usual perspective, or emotions shifting within you that aren't really yours. If you're using Fitzpatrick's technique, your whole imaginary dog-body will be empathic with your pet dog's-- so take note of what you physically feel...if your dog's paw is hurt, for example, your hand will be aching.
Sometimes, you might need to be the first to talk... to invite them to converse with you. Mundanely copying body language, or a certain tone of voice, and command words or their name is definitely something animals can understand-- but I haven't experienced them really speaking English or other human tongue. So if you can, try sending out your message (down the link of light, or in the part of your mind that you're receiving... that last one's hard to explain, but I think it's more effective) in images and emotions.
Hope this helped!