First, don't open his carrier before you've closed all doors and windows. Until your home feels familiar to him, his impulse may be to escape. Cats are ingenious at squeezing through small openings. Isolate him to start with. Set up a spare room with a litter pan, food, water and a bed. He may choose to sleep in his carrier since that has his scent on it, or he may scoot under a bed or couch until he feels safer. But unless there's bedlam in the house, it won't be long before curiosity gets the better of him and he's pattering around his room, sniffing everything.
You'll see him pause and gape with his mouth ajar. This is the Flehmen response and it tells you he's analyzing the scents of your home. His new environment becomes less intimidating with each sniff. If you have another pet, he'll smell that too. They don't need to see the whole house and family to find out all about them.
Your resident pet will probably be on the other side of the door doing some sniffing of his own. Let them 'meet' this way first, sniffing under the door. Eventually you can open the door and let them see each other. That may take a few hours or a few days. Each animal is unique; judge when everybody seems relaxed enough to move to the next step.
Pets are like children in that you'll see some sibling rivalry if you devote all your attention to the new pet. Remember that your other pet is watching and will probably feel jealous. Give special attention to your original pet as often as to the new one.
Once the door to your new pet's room is open, your other pet will want to assert his ownership by going in and using the litter pan or eating the food. You can minimize the territory issue if you put something that smells of each pet, like bedding, in the other pet's area. It will be sniffed intensely and treated with suspicion at first but will soon become part of the background. When they enter each other's area, they'll be reassured to find their own scent there.
With enough space in the house to avoid each other if they wish, your pets will soon be nonchalant about each other at worst and may even become friends. But if you have a couple of possessive or high-strung animals, you may need more time to assimilate the new one. Try giving them treats when they're together -- not so close that they're too intimidated to eat but close enough that they associate a pleasant experience with each other. Once the treat is finished, separate them again for a while. If you have a dog, keep him leashed while you introduce the two pets. Let them set the pace of the introductions.
Warn your kids to go slowly with the new pet at first. Even a laid back, gentle animal will feel overwhelmed by too many new stimuli and may lash out if he has too much to deal with. Remember, his behaviour during the first days won't be typical behaviour. He will become more relaxed with each day and his personality will emerge more and more.
Try to give your new pet the kind of diet he's accustomed to at first. You can gradually mix in increasing quantities of the food you plan to keep him on. But the slow introduction of the new diet will help prevent digestive upset. If you can keep stressors to a minimum by introducing him to your home slowly and by maintaining his former diet, you'll minimize the likelihood of problems such as diarrhea from change in diet or a cold that can leave a cat sneezing and stuffed up if stress lowered his resistance.
Remember above all to wait at least a month or two for everyone to settle down. In many cases, the adjustment may be complete in a week or even overnight. But if a timid or sensitive animal needs longer, you'll find he's worth waiting for when the day comes that he climbs into your lap and nuzzles you because you belong to him.