**disclaimer: this is my opinion/based on personal experience. I'm in no way speaking for or on behalf of the shelter**
The first and most important fact to keep in mind is that your new pet is seriously stressed.
No matter how well we treat the animals at the shelter, the shelter itself is a stressful environment. There are new sounds and smells all over the place and the animals are prevented from doing many of their natural behaviors in response. Cats often want to hide while they adjust and figure things out, but there's really nowhere for them to go in a kennel. Dogs like to thoroughly smell things to gather information, but they are separated from everything by doors.Imagine being in a crowded room with everyone talking at once as they are constantly moving. It would quickly get under your skin, especially if you were trying to find someone or trying to understand what someone was saying. Now imagine living in that room.
If you think about it that way, it's amazing shelter pets do as well as they do.
Your home is not their home.
Not yet, anyway. As overwhelming as the shelter is, it's familiar. You're taking them out of that into a brand new environment. They don't know you, they don't know this new place, and they don't know the other two- and four-legged members of their new family.
I'm sure your home is awesome. And if you give your newly adopted pet a chance, they're likely to think so, too. But you have to get them through those first few hours (days, weeks...) and start convincing them that you're someone to be trusted.
Cats will probably hide. Let them - you do more harm than good by dragging them out and forcing them to confront a situation that has them scared out of their minds. Dogs may chew or have accidents. Rather than punish them and start your relationship off with anger and pain and fear, gently redirect them to appropriate chew toys or the correct place to potty.
Even if you're familiar with animals, listen to the adoption counselor and read the provided materials.
This person is familiar with shelter pet behavior and may even be familiar with the animal you are adopting. Training and introduction procedures are constantly being revised, so information you read three years ago may be completely out of date now.
Just like people, animals have their own personalities. As a personal example, adopting my first two dogs (nine years apart) went smoothly. They got along with people and other animals and there were only a couple of minor chewing incidents as they got adjusted. The dog I adopted last August is a completely different story. She was fearful and anxious and snapped at friends and family. My previous adoption experiences did nothing to prepare me for this one.
Just like people, animals have their own personalities. As a personal example, adopting my first two dogs (nine years apart) went smoothly. They got along with people and other animals and there were only a couple of minor chewing incidents as they got adjusted. The dog I adopted last August is a completely different story. She was fearful and anxious and snapped at friends and family. My previous adoption experiences did nothing to prepare me for this one.
I know you're excited, but please be patient with your new fur buddy.
Commit to giving your animal time to adjust. Try not to get too disappointed when your new cat won't come out from under the couch for the first day or so. 'Puppy proof' your home, even if you're adopting an older dog. Keep important items out of reach so they can't be destroyed.
In a few weeks or months, you won't even remember life without your new pet. All of the time and effort you gave while helping your critter get acclimated will be rewarded when your dog nearly wags his butt off when he sees you or your cat makes biscuits on your lap, purring contentedly as you scratch behind her ears.
It just takes time. And patience. And love.