If you already have other pets the most important pet in the world must be the animal(s) in your home already. That animal must feel secure in its surroundings and its place in the family. It must also not feel like its territory is being invaded. Your old pet will be very defensive if the new animal is just plopped into the middle of its house and given the run of everything. To avoid conflicts there are several steps that can be taken to help.

1. Introduce the new pet in a neutral area. A friend's house or a park, are good choices. Ideally the introductions should be done before the new pet is chosen in case they absolutely despise each other. This process usually works better with dogs than cats.
2. The new pet should be limited to only one area/room of the house for a few days. This should not be your other pet's favorite room and should not cut them off from access to their regular feeding area and/or litter box or sleeping area. This lets the old pet know they are still king of the house and sets up limits and security for the new pet. The old pet will also be able to get used to the smell of the new pet before actually letting it out into the rest of the house.
3. You may want to have a separate play/cuddle time for a few days to a week where the old pet doesn't encounter you spending time with the new pet. You're still giving plenty of attention to the new pet but the old pet doesn't feel displaced.
4. When its time to introduce them to each other in the house make sure to pay all your attention to the previous pet and/or children. Ignore the new pet. This lets the new pet know who gets attention first and sets up a good routine.
5. For the first several days, if not weeks, the new pet should only play with other pets and children with adult supervision. This will let you be sure things do not get out of hand and continue to reinforce the routine in the house.

If you don't have other pets but do have children the same approach should still be taken, especially with younger children. While a child won't necessarily feel displaced in the family, the new pet needs to know that the children are going to get attention first. This will help set up a routine with the children. After a week or two the children should be feeding the pet, with supervision. If they are old enough they can also help with grooming and walking tasks.

It is also important that the children understand the rules of having a pet in the house. Pets are not toys; they are living creatures that can be hurt by rough physical behavior. They may not like being poked, having their hair pulled, or being teased any more than the child would. Also loud noises and surprises don't usually mix well with animals. Establish and enforce the rules with your children from the beginning to avoid any unfortunate incidences. Teach other good habits like always allowing the pet to come to you, don't chase. Pet the animal from the collar back, moving slowly with the direction of the fur.

A few other helpful suggestions are to begin handling all parts of the pet early on. Play with feet, ears, tail, mouth, etc. This will get the animal used to being handled. For children handling the pet, vet visits and perhaps at other times when the animal may be stressed it is important the pet feel comfortable with being touched. Practice taking things away from the dog. Make the dog get up and move out of your way instead of going around it.