Living with cats and dogs is not as easy as it sounds, but with some excellent management, some proactive training, and some time, you can find harmony.
Puppies and cats
Puppies are playful, innocent, excitable little creatures that lack impulse control, have the best intentions, and can't read signals just yet. This is what makes them so challenging around cats when you bring them home.
New rescues and cats
New rescues of any age generally have a honeymoon period of about 3 weeks to 3 months, depending on their age and their history. While many rescues do their best to vet these dogs with various other animals, there is no guarantee that a "cat-friendly dog" will be friendly with your cats. Go in with eyes wide open and know that you might be in for a lot of management and work.
Ideally your cats can have the run of the place while they're most active (night) and the dog has the run of the place during the most active hours (day).
Baby gates are your friends. Use them to separate spaces so that the cats can escape. Some gates are taller than others (great for larger or acrobatic dogs) and others have a cat door in them (great when you have a dog who doesn't fit through that door, OR a cat who is older and unable to jump over the gate).
If you cannot manage with gates due to an open-concept home, you can utilise a LTCA to protect your free-roaming cats from your puppy or dog whether you have an open-concept home or not!
If management is still difficult, have your dog on leash when the cats are around and active. Be armed with high-value treats for your dog and always ensure that your cats have a safe and easy escape route.
Every time your dog chases your cat, it's a hit of dopamine similar to how you might feel if you hit the jackpot at the casino. It's very reinforcing and a long history of rehearsing this behaviour is difficult to reverse. Prevent the chase. Every time.
Train what you want to see
Spend some time (sans-cat), training your puppy or dog what you'd like them to do when the cat is (eventually) around. Here are the key behaviours your dog needs in order to be successful...in the order of importance!
FIND IT! To help your dog visually disengage for a moment
LOOK AT THAT! To help your dog look at the cat and reorient to you immediately and automatically
GO TO MAT! To give your dog a place that is magnetic and highly reinforcing to hang out on
DOWN! You can't chase a cat if you're lying down - this is called a DRI - Differential Reinforcement of an Incompatible behaviour. Supernerd!
STAY! Not necessarily the most important, but down the road, maybe so!
RECALL! Always a life-saver. Potentially for your cats one day, too!
Avoid punishment at all costs
Punishment creates what we call a "silent biter" - a dog who doesn't warn us before they bite. This applies to situations with cats as well. If we suppress their natural urge to chase small prey-like animals, it doesn't go away, it's simply hidden below the surface.
Rather than punishing your dog, go back to management and tighten up.
Never leave them unattended together unless you have a long, long track record of no issues. This means 6+ months of zero chases, zero intensity, zero anything.
Never let your cat teach your dog a lesson with their claws or a bite. This can not only cause serious physical and emotional damage to your dog, including infection, but it can instigate a fight that your cat cannot win.