How to Introduce Your New Puppy to Your Other Pets (dog or Cat)

First, if at all possible have your older dog meet your new puppy on neutral ground, like a park, the dog park or even a parking lot. This makes things so that your existing dog will not perceive the puppy as intruding on his territory.
Once they have been introduced on neutral ground and if things go well, then you can walk the two together to your home or the car.

If your older dog is aggressive to the new puppy then you will need to go on to step 2

Put your family pet into another room or the garage, then bring in the new puppy and place him into a dog crate with a water bowl or bottle clipped to the inside of the crate. Make sure that your new puppy is wearing a collar and you have a leash for him.
The crate should be of the type that is made out of high density plastic with a wire door.
This does 2 things;
1. gives your new puppy the feeling of security by having his own "den"
2. affords greater protection for your new puppy from unwanted attention or aggression by your existing pets.
Place the leash on or very near the puppy's crate so that it is handy when you need it to take the puppy out to go potty.

Now that the puppy is in his crate you can let your existing pets out of their seclusion on leash to go back into the house and have access to the room that the new puppy is in.

Once you and your dog or cat is in the room with the puppy you will spend time brushing or playing with your current pets. Totally ignore the puppy in the crate.

Once your other pets are calm then they can go to "visit" the new puppy while the puppy is still in the crate.
Some puppy's will bark and growl fiercely at your pets, ignore this behavior, do NOT talk to the new puppy but DO talk in soothing tones to your existing pets unless they are showing signs of aggression.

If your pets seem friendly and want to engage playtime with the new puppy do not allow it for at least 2 days. This is to give the new puppy time to feel secure in his new home and to give your current pets more time to feel more comfortable with the new puppy.
Puppy's often harass older dogs in an attempt to get them to play, this can cause an older dog to growl, snap and or bite the new puppy. As long as your pets are of a stable temperament the discussion between the two should stop there and the puppy will learn his boundaries.

In the mean time when you decide to play with your puppy you should have your other pets out of the room to avoid jealously issues.
When you take the puppy out of the crate to go outside you need to be doubly careful to insure that the puppy will be safe because this area will be perceived by your other pets as THEIR territory and it may take a little while before things settle down.

After about a week of the puppy spending much of his time in the crate you will have accomplished 2 things.
1. your puppy will be well on his way to housebreaking if not totally housebroken.
2. you will have given your existing pets time to accept the sight, smell and presences of the new puppy.

I still would not allow the new puppy to be with an older dog without your supervision for a couple of weeks if all goes well or a couple of months if your current pets remain hostile to the puppy.

Some dogs especially may never accept a new puppy into their household. These dogs are usually dominant dogs with little training. They see the house as their exclusive territory and will defend you and it from the puppy. In cases like this you may be able to get the older dog to accept the puppy in your presences but can never be trusted if you are not there to intervene.

With patience and understanding you can make this work. An older dog usually will accept a puppy as long as the introductions and time is tailored to the older dogs needs. He was there first and should receive respect from the new puppy, this is what pack order is all about.