How do I introduce my new pet to other family pets?  It depends (sorry, I know that's not what you're looking for). First, understand the Rule of Three: It's going to take your new pet about three days to decompress/get used to the new environment; three weeks to learn the household routine; and three months before they are well and truly settled.  Here are some basic guidelines that will help that process along:
 

1.  Understand that not everyone is going to welcome the newbie happily.  We may ask you to bring a resident pet in to meet the prospective adoptee, but understand that it's a very different environment at home than at the shelter our out in public. Everyone--partners, kids, friends, and other pets--will need to be willing to make some adjustments as you're sorting out the new dynamic.
2.  Provide supervised opportunities to become familiar with one another. For cats, one of the best ways is to have a separate room for the new cat, complete with food, water, and litter box. Let the resident cat(s) and/or dog(s) become familiar with the scent and sounds of your new feline friend. It's okay if they sniff on each side of the door. You'll go in to spend time with your new friend--often, we hope--and let the other pets smell you when you come out of the room. After a day or two, you may try placing a sturdy pet gate across the doorway so that the newbie and resident pet(s) see and smell one another more, but the newbie still has a safe place if s/he is feeling stressed.  When those on both sides of the gate seem comfortable, allow supervised interactions. Initially, those should be brief so that everyone understand where they fit in the pecking order and that you are in charge.

The approach with dogs will be a little different. Try to have the first meeting between the new pup and resident pup(s) on neutral territory: a park, out in the front yard, or some place where the resident dog(s) has not claimed as his or her territory. Have a friend or partner control the resident dog on one leash while you have secure control of the new dog with a different leash.  Let them sniff (butt sniffing is dog for shaking hands), and then take them on a short walk together. Immediately correct any mounting or aggression and separate the dogs.  In the house, keep the new dog on a trailing leash at least for the first few days. This gives you an opportunity for control should something unexpected happen. If you have any concerns about possible fights have trailing leashes on the resident dogs, too. Lastly, supervise, supervise, supervise. If you turn them loose in the backyard without being there yourself something may go south, and you don't want that. In the best cases, the new dog is welcomed as a new pack member immediately, however, understand that it's a process.

3. Manage your expectations. The initial week may be rocky, but that doesn't mean the adoption was a bad decision. New beginnings are stressful for pets and people; offer a bit of grace to all involved. There may be potty accidents from both the resident pets and the newbies. Resident pets may not be thrilled initially to have to share you, space, and toys. With consistency on your part, all can learn to live together peacefully, just understand that it's a process.