If your dog is wearing a life vest, attach a ski slope leash to it. Attach the other end to an inflatable tube you can relax in, while the dog pulls you along. This is called towing, and is a valuable rescue skill for your dog to learn. If anything were to happen out on the water, your dog would suddenly be equipped to pull you out to safety. But it goes without saying, be logical before practicing this. If your dog is very small, this may not be the ideal activity.
This Hawaiian sport is key to building better balance and muscle tone for you and your dog. Since most paddle boards (think of an enlarged surfboard) are built with two people in mind, you can get in the water, hop onto the board, balance yourself and start moving with the paddle. Your dog will undoubtedly get in the water with you and follow along. Call the dog up and cheer when there's successful boarding. From there, let your dog stand or lay down on the front end. All you need to do is paddle right along as you normally would.