Summarize this article:

Not all dogs show outward signs of a roundworm infection. If you have a puppy that is showing clinical signs, he will have a potbelly. In addition, your puppy’s coat may look dull, and he may not be growing and putting on weight as he should. He will be smaller than what he should be for his age. The potbelly appearance is due the large number of adult roundworms in your dog's belly. When roundworms find their way to the intestine, they feed off of partially digested food. This robs your dog of vital nutrients and causes him to experience digestive upset, including vomiting and chronic diarrhea. You may be able to see worms in the vomit.  Puppies that are four to six months old and have heavy roundworm infections may vomit large worms that are still alive, which could be very distressing to you. The roundworms will be an off white color. With early diagnosis and proper treatment, it is unlikely for your puppy to have such a severe infection. The vomiting is due to the adult roundworms irritating the lining of your dog’s stomach. Call your veterinarian if you see live worms and are unsure how to dispose of them. The chronic diarrhea may cause your dog to eliminate more often in the house, which would leave worm eggs in your home and potentially expose other pets, or re-expose him, to infection. With the digestive upset, your dog will eat less. In the complicated life cycle of roundworms, the larvae will migrate up to your dog's lungs, which will cause him to cough. The coughing action moves the larvae into your dog's throat, causing your dog to swallow the larvae. If your dog has a severe roundworm infestation, the larvae in the lungs could give your dog pneumonia. Signs of pneumonia include difficulty breathing, a deep cough, and lethargy. Looking at your dog’s feces isn’t exactly enjoyable, but it could help you determine if your dog has roundworms. Adult roundworms are three to four inches long and look like spaghetti. The larvae would likely be too small for you to see with the naked eye, and the eggs would not be detectable at all.  If you see adult worms, you may want to place a small sample in a tightly sealed plastic bag and take it with you when you take your dog to the veterinarian. Wear gloves when you collect the sample and wash your hands thoroughly afterwards. Roundworms are usually found in the feces.

Summary:
Look at your dog’s appearance. Identify signs of digestive upset. Look for signs of respiratory problems. Examine your dog’s feces.