Many people adopt a pet then want to return it after only a few days because it doesn't get along with other pets in the family. New pets must be carefully introduced to others in the home.
Pets and kids: Children younger than 4 probably shouldn't be involved in being responsible for a pet. Ensure that your children understand the consequences of playing roughly with a pet. A cat can leap out of harm's way, but a dog may be more vulnerable. Teach and reward gentle treatment.
Dogs and dogs: Adult dogs usually accept puppies because they don't see them as a threat. Mixing two adult dogs is trickier. Make the introductions in neutral territory like a park or playground or keep the new dog in a gated indoor pen and offer supervised "visits" with your current dog. Use treats to reward calm interaction.
Cats and cats: Kittens will often try to test the limit of adult cats, so increase their exposure gradually and use treats to reinforce positive interaction. Also realize that kittens need some privacy - somewhere to hide, like a covered bed or a safe place to call their own.
Dogs and cats: Be sure to teach your dog to respect the cat's boundaries. When you make the introductions, keep your dog leashed and make sure the cat has an escape path. If your dog reacts calmly, you can eventually drop the leash, but be ready to grab it if he makes a move toward the cat.
Most pets can learn to get along but don't rush it. It takes time and patience.