So if a very young puppy is coming into your house that already has a dog or a cat in it, it can be a little intimidating or the dog can just rush right up without any social skills and kind of harass the cat or dog. So you want to make sure that your dog our your cat that's already living in your house is comfortable with the dog before the dog is in their house, in their stuff, kind of, you know it can be a little stressful for them as well.
Usually I suggest with dogs is that we introduce a puppy to an existing dog outside of the house where they can kind of sniff and explore each other a little bit before your puppy's running around on your existing dogs bed and in his toys and everything else where the dog might get a little uncomfortable.
You'll know your dog well enough to know whether or not he'll accept the dog socially or not in the house right away. But even with dogs who are really cool, I usually suggest that they meet outside of the house someplace where they can be calm. And then also, I would have the puppy in an enclosed space such as this in a long term confinement area like an exercise pen with something to chew to keep his attention while your dog comes and kind of explores him a little bit.
Especially if the puppy isn't fully vaccinated, you want to be sure not to bring him outside to a high risk area so you can also utilize the exercise pen with something to chew so that your adult dog who already lives in your house can kind of explore the puppy, get some olfactory information from him before it's a lot of face to face confrontation so they can kind of get used to each other. Then they can come out and explore and play a little bit.
As far as cats go, it depends on how social your cat is and how comfortable your cat would be with dogs but a lot of times cats can get away a lot faster than puppies can catch them and that's a good plus. But also, the food and litter box can be in a different space or a different floor or someplace where the puppy can't get into them and the cat can have some space to themselves just so that the interaction and the meet and greet can be extended over time and go with the cats comfort level as opposed to just bum rushing them and insisting that they get along right away.
That's how I would do it and that's how you introduce your puppy to an existing pet.