Make the Introductions
A new cat should be introduced carefully into the household. Those first few days and weeks can make all the difference in how the cat relates to the family in the long run.
Have your cat's basic supplies -- litter box, food, dishes, scratching post, bed -- ready before the cat arrives. Whether you're bringing home a kitten or an older cat, make sure she doesn't get frightened and disappear. Close windows and cut access to spaces where a cat might hide. Try keeping Kitty confined with a litter box in a small space like a bathroom. Take the cat carrier into the bathroom, close the bathroom door, set the carrier near the litter box, open the carrier gently and let kitty venture out. Once Kitty has settled down and feels safe in the bathroom, open the door and let her venture out. Keep the doors to most other rooms closed at first. Let Kitty explore the house gradually.
Have a veterinarian check out your new arrival before bringing it home. If you already have a cat, you may need to leave the new kitty in her bathroom safe house for several days. The resident cat(s) will start to meet the newbie by sniffing under the door. As long as you don't let an active kitten annoy an adult cat too much, the adult will probably accept the kitten eventually.
If you have a dog, keep it on a leash until the dog and your kitty are used to each other. Don't let it chase or bark at her.