Listen Up, PLEASE!
This protocol is the result of CENTURIES of experience by many expert cat-keepers. If you have been given this sheet, there is a good chance you may not have had quite this much experience with the mind of the cat, so PLEASE follow it carefully. In my own 35 years of dealing with the feline psyche, I have known this method to fail only once... and have witnessed countless cases of failure caused by a human's impatience or decision to disregard the needs of his kitty companions!
The new cat must be set up in a separate room with his own food, water, litter pan, etc. If possible, this should be a spare room, fairly quiet and as free from household traffic as possible. PLEASE do not choose your other cat's favorite room or your bedroom if your first cat is used to sleeping with you. Spend as much time as you can in this room with the new family member, allowing him to get accustomed to the sounds and smells of his new home as well as to you, his new person. This phase should continue until kitty is completely comfortable in his new surroundings. Whether this takes 3 days or several weeks is dependent upon the individual cat's level of self confidence. Wait it out!
This is SWAP TIME and the most important phase. Now it is time for your new kitty to get to know the rest of your house and for the resident cat/s to learn more about the interloper who's been in the closed room. You do NOT want them to see each other yet! You may need to use a carrier. Get the new cat out of his room and put your other cats into it, close the door and let the newbie out into the rest of your home. This allows the resident cats to snoop all they want without coming face to face with the intruder and gives the new cat the chance to find all the hidey spots, scope out his new home and absorb all the "other cat" scents without the threat of confrontation. Do this for an hour at a time, two or three times a day. Then everybody goes back to their Step 1 locations for a rest. Phase 2 is ONLY over when all parties are camped out on their respective sides of the door to the newbie's room and playing "footsies" underneath the door. This could take anywhere from two days to two months, again depending upon the confidence levels of all purrsons involved. Pushing the cats into Step 3 before THEY are ready will either cause a scene that will take you MONTHS more to "undo" or result in failure to adjust which just ends in heartbreak for all concerned!
When they are playing with each others' paws, they will have accepted the "others" presence well enough to not feel threatened and are now ready to meet... under supervision, of course! There will still be some posturing and hissing as they work out their relationships, but if you've done well, there should be no "trouble." After 20-30 minutes of visiting or at any sign of stress or chasing, put everybody back for a rest You may increase the "visit" times daily as long as everyone remains relaxed. Do not leave the whole crew alone together when you can't supervise until there are no lingering signs of stress or unease.
Good Luck and Enjoy Your New Family Member!