There are several steps that every new cat owner should take when introducing the new cat to the resident dog. A pet owner cannot simply throw the cat and dog into the same room and expect harmony; the dog-cat introduction process will take several days, even as long as several weeks.
Create a "safe room" for the cat. Place the cat inside a "safe room" that can be closed off to the dog. A spare bedroom is ideal for this purpose. This safe room will be a place where the cat can feel comfortable and safe; as the dog-cat introduction process progresses, this room will be somewhere the cat can go to be away from the dog if the need arises.
Bring the dog to the door of the cat's safe room. During the first few days, the door to the cat's room will remain closed. Bring the dog to the cat's door and allow him to become acquainted with the cat's scent. Reward good behaviors on the dog's part with treats or praise; discourage negative behaviors like barking with negative reinforcement like verbal correction or a spritz from a water bottle.
Perform a scent exchange. Place a towel or blanket in the cat's room for a day or two. Place a towel or blanket in the dog's bed or favorite sitting area for a couple days as well. Then, switch the towels or blanket; giving the cat the dog's blanket and vice versa. This allows the dog and cat to become familiarized with the other animal's scent.
Allow supervised visitation with the dog and the new cat. Once the dog's excitement about the cat has worn off (usually after a couple of days), and once the cat has become familiar and comfortable with his new owners and with his new safe room, it's time to allow the dog and cat to spend time in the same room. Two people will be required for this exercise, which should be performed several times a day. One person must supervise the dog, while another person must supervise the cat. Sit with the dog and the new cat in the same room. Begin by sitting on opposite sides of the room and reward calm behaviors in the cat and in the dog. If the dog begins to bark or get excited, he should be given negative reinforcement like a verbal correction. The key is to promote peaceful co-existence. As the dog and cat become more comfortable, lessen the distance between the dog and cat until they are sitting side by side.
Allow the dog and cat to interact. Once the dog and cat are peacefully co-existing and they're comfortable sitting side by side, allow the dog and cat to interact. These interactions must be supervised, especially in the beginning. Be ready to separate the dog and the new cat if signs of agitation or fear are observed (i.e. hair raised on the cat's back).
Allow unsupervised contact between the dog and cat, but maintain the cat's safe room. Place a baby gate (or two if the dog is large and able to jump over one baby gate) at the doorway to the cat's safe room. Leave the door open and allow the cat and dog to interact through the baby gate. As the cat feels more comfortable with the dog, he will venture out of his safe room. Always leave the cat's safe room open and accessible. With time, the cat and dog will learn to peacefully co-exist.