When you bring your new friend home . . . .
Your dog has been under stress. He has been in a strange place full of other animals. Now he is in a new home, with a new family and maybe even other pets. Although everyone will want to meet and play with him, keep the initial interactions to a minimum. Let him explore and investigate his new home and yard.
Use caution when introducing pets to each other. Some may eventually become friends and others will just co-exist. Before introducing your new dog to pet(s) already in your home, take the dog for a brisk walk to release energy. Then allow him to sniff where the other pet has been. Introduce slowly and keep other dogs on a leash.
When introducting a dog and a cat, When the dog is in a calm state and still on a leash, allow the dog and cat to view each other at a distance. You should exude calm but also express firmness and not allow the dog to chase the cat or the cat to scratch the dog. Another human family member should hold the dog leash while you calmly pet the cat, thereby letting the dog know that the cat is part of the family. Most dogs want to please their human companion. If the dog remains fairly calm, allow the dog, while still on a leash, to get close to the cat.
A dog's basic instinct is to chase a cat. Whether the dog chases or not depends on the introduction and on the cat. A cat who is not afraid of dogs and does not run is less likely to provoke a chase. The new dog and your cat should not be alone together for a few weeks the dog may still chase a running cat. Also, if your dog sees the cat outside, he may feel like the cat is fair game for a chase.
It is important to give equal attention to your original pet. Do not ignore him as this may cause resentment of the new pet, and most important is that the original pet too needs to feel loved, even more so because of the new addition to your family.