Introduce the new pet gradually
First introductions should be with the new cat or dog in a crate or carrier. Let them meet each other in a safe and non threatening environment. Your dog or cat will not view the new family member as an intrusion if allowed to meet them on their own terms. Once they seem comfortable with each other open the crate and let them have a face to face, supervised, meeting. Don't leave them alone together until you are sure everyone will be safe and happy.
Beware of your emotions
Your cat or dog will be able to sense if you are apprehensive of them meeting the new addition. Keep a no-nonsense, upbeat attitude and don't fuss over their meeting. During this time give your pet extra love and attention to let them know their place is the family is not in jeopardy.
Keep things consistent
Don't change sleeping and eating areas or schedules. Let your furry friend know that, although there is a new member of the family, their life will not significantly change. Keep everything equal between the new siblings, they will know the difference!
Know your pet's personality and inbred instincts
Depending on the breed, your pup may react certain ways out of instinct. My Sheltie, a herding dog, instinctively herds anything that moves and the cats in the house certainly qualify. Cats will either tolerate this behavior, find it annoying and fight back, or shy away. To keep kitty happy, pup safe, and harmony in the household, it is best to have high places where kitty can get out of the way! A perch by the window, cat tree, or similar high cat friendly place is perfect when kitty wants some time alone.
Give each pet a place of their own
Everyone needs a place to call their own and pets are no exception. A crate or dog bed is perfect when your pup wants some quiet time alone. A high window perch, cat tree or other high place accessible only to kitty, will keep them happy and content. At meal times, feed kitty siblings together but give each a separate bowl, giving them the chance to share if they choose to. When cats and dogs are together, be sure to feed kitty on an elevated area out of pups reach.
The new siblings may be immediate friends or it may be a process that take some time. Allowing them to get to know each other at their own pace and on their own time is the best way to assure success. You may soon discover that they are inseparable companions.