Some pets will never become friends, but may merely co-exist (some may even be a danger to the other pet). At the very least, it takes time for pets to adjust, especially for the resident pet to allow another animal into its territory. Do not ignore your original pet! he needs to feel loved.

To introduce the cats, keep them separated for a few days, allowing them to smell each others bedding. Rotate the bedding to help them adjust to the new smell. Let the cats sniff at each other under a door. After awhile, let the cats meet. There may be a lot of hissing. While the first encounter may seem hostile, allow the cats to work it out on their own. It will go faster that way. If the cats start to fight, do not separate them by hand. Spray them with water, and then close the door. Try again a few days later.
When your cat meets your dog, introduce them in much the same manner as with cats. keep your dog on a leash. Do not allow your dog to chase your cat. Supervise all initial meetings.

Pay the older cats more attention than you do the newcomer. Make a special effort to pat them and play with them more. Therefore they won't feel as threatened by the younger cat.

Keep their food dishes in separate locations (preferably on opposite sides of the house) until they get to like each other better. Animals get especially territorial and neurotic about their food, so any tension between the cats will escalate at feeding times. Feed the older cats first.

Accept that a certain amount of hissing and chasing is inevitable for the first couple of months. Keep spray bottles (get clean, empty plastic ones at the drugstore really cheap - don't reuse bottles from beauty aid or household products because they can leach into the plastic and taint the water) filled with water around so you can spray the cats if they really start to hurt each other. Spraying them with water will break up even the most intense fight.

Sometimes they just have to establish the hierarchy. Until they figure out which one is the dominant one and the remaining chain of command there isn't too much to do except the water bottle trick to make sure they don't kill each other.

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