New Dogs and Existing Dogs:

• Neutral Territory

Dogs should always be introduced to each other on neutral territory.    Otherwise, the new dog will always be seen as an "intruder" and react defensively.  Dogs need about three seconds to make a "judgement call" about another dog.  After this, either hackles go up, or tails wag.   After the initial impression, dogs introduced off lead (in a fenced area) generally do better than those held on tight leads because they can interact without worrying about their owner.

• Male or Female, Confident or Submissive

Many times (but not always), getting a male dog when you already have a female dog or a submissive dog when you already have a confident one (and vise versa in each case) works out well.  It is a common misconception that littermates will always get along.   Littermates, especially those of the same sex, probably argue as often as human siblings argue.  Dogs that are spayed and neutered will have much greater chances of getting along with each other, since hormones won't be involved. 

• Matching Ages

Another common misconception is that it is a great idea to get two puppies at once.   Two young puppies can actually be harder to train because it will be harder to get each to focus on you.  Young puppies can learn many valuable lessons from older, more settled dogs.  Make sure the new puppy sleeps in his crate when you are not home and at night; it will give your older dog some rest from all of his "teaching".  

• Individual Space and Time

Like humans, it will be important for each dog to have some space.  Each dog should have its own crate and should be fed there.  Free-feeding (allowing bowls of food to sit out in the open all day) will probably invite food aggression problems.   Each dog should be trained separately and should get some "one-on-one" time with its owner.  Special attention should be paid to give the dogs constructive activities to do to keep them busy; otherwise, they may team up to get into trouble.  
 

• Pack Dynamics

Any more than two dogs is a "pack".  The dogs in your "pack" will have to work out their own hierarchy, even if they all (as they should) consider you the ultimate "alpha dog".  You should not try to influence these interactions -- helping out the submissive dog or putting down the dominant dog --   unless the behavior gets aggressive.  Normal "wrestling" (mounting, tackling, ambushing) will occur daily in a pack. 

• If At First You Don't Succeed...

Like people, some dogs just don't like each other for whatever reasons.  There are some chemistries that just don't blend well together.  Just because your dog doesn't like one dog, doesn't necessarily mean he won't like all dogs. 

New Dogs and Existing Cats:

 

• The Scent Exchange

It is probably easier to match an older cat with a younger dog and vise versa.  If a dog or cat has been around cats or dogs before, it will probably generalize its experience.