The sooner you can introduce a new puppy to existing cats, the better. A young puppy’s personality and behaviour are more of a blank slate, so begin your introductions to other household pets during their first week at home.
Start with scent transference, swapping a piece of bedding from each animal to familiarise them with the other’s smell. Next, spend a few minutes playing with the puppy alone, sitting on the floor so your lap and hands become laden with the puppy’s scent. Then get your cat's favourite treats and start interacting with the cat while the puppy is present.
Close the doors but make sure the puppy can’t jump on or chase the cat, even though they might be eager to play! Keep a lead on your puppy but let the cat walk around freely. If everything goes well, you can use treats to encourage your cat towards the puppy.
The learning is more on the puppy’s side than the cat’s. If the cat is moderately confident, they may be content to be close to the puppy from the first session. Most puppies and cats will be fine in each other’s company, but always supervise them until you’re certain they can be trusted together. Some dog breeds are poorly suited to living with cats; breeds such as Greyhounds and Beagles are bred to hunt smaller animals and their instinctive behaviours, such as chasing, are likely to distress your cat (and other smaller pets).