INTRODUCING THE DOGS
It’s even more important to go slowly when introducing dogs. As Dr. Suzanne Hetts says, “Dogs can kill a cat very easily, even if they’re only playing.” Of course this is even more true when introducing a kitten.
Some cats have never been around dogs or have already had a bad experience with them–that fear response is what they remember. So if a cat has already learned to be afraid of dogs, you’ll probably have to go even more slowly.
Even a cat that has grown up with dogs and overcome its instinctual fear of a predator (usually) several times its size won’t necessarily generalize that your new dog isn’t something to fear. And of course, it doesn’t matter if your dog is comfortable around the new cat–it’s the new cat’s response that’s most important in these introductions. If it shows fear, you’ll have to back up and work with that.
The new cat will probably have to learn that this huge predator it now lives with is safe. Include the dog in the eating and scent-switching techniques, above. Then pick a room with you, a friend, the dog and the cat. To keep the cat’s fear level down, you’ll need to make sure the dog isn’t making big movements or noises. Leash the dog and put it in a Down Stay, if it’s had any training. If not, leash and keep it calm.
Start with the home dog and new cat on opposite sides of the room, each with a person giving treats (use clicker play if the dog knows how to do that) so they both associate good things with the new animal. Build up exposure to each other with many short visits so neither animal gets antsy.
Once the cat is comfortable, allow it to approach the dog at its own pace, keeping the dog in the Down Stay. If the cat gets fearful or aggressive, you’re going too fast. Go back a couple of steps and start over. Repeat this process in small steps–probably over days–until they are clearly comfortable with each other.
I hear people say “My new cat attacked my dog.” If you let the dog get close enough for the cat to go into attack mode, you have already pushed the cat way past its fear threshold. These pictures show a cat’s body language from beginning fear to fighting for its life:

This cat is just beginning to feel threatened, with the dog starting to focus on it and the cat freezing with its ears down. (thanks to James for the picture)

This was probably set up to be amusing, but if you really study it, you’ll see the kitten is afraid and the dog could attack at any moment, especially if it perceives the kitten as challenging its food.

This cat is trying to make itself bigger to fend off a dog that is already barking and challenging it, the same technique that wildlife officials recommend to humans encountering bears. (thanks to Tom for the picture)

The owner has allowed the dog to get too close too soon, forcing the kitten–probably 1/10th its size–into attack mode and entrenching its fear response to dogs.

This terrified kitten–probably only a couple of months old–is using every technique it knows to fend off this pup: arched back, fluffed fur, vocalization. Even though the dog is a pup, you can see the big difference in size.

This dog is in near-full attack mode and the cat is fighting for its life here, with its only advantage being slightly higher than the dog.
Always be sure the new cat has an escape route and can get to its safe room, and keep your dog leashed until you’re sure the new cat is no longer fearful. Of course if you have more than one dog, introduce them singly just like the cats.
It goes without saying that actively punishing any of the animals for being aggressive will link the new cat to that fearful response in everyone, setting up everyone for failure.
You may think that taking couple of weeks to introduce animals is time-intensive. Remember, that’s nothing compared to the time and emotion you’ll have to invest down the road, undoing fear that has turned into aggression.
So to recap, if you’re bringing home a new kitty:
Change one thing at a time.
Go slowly.
Seriously.
Really slowly.