The pet you see in a shelter may act completely different in a home environment. Allow lots of time to get your adoptable used to their new home.
For dogs, introduce them to their new surroundings in a calm manner and avoid taking them on adventures outside the home straight away. It's not a house inspection, so you don't need to rush them around every corner of the house, you can keep them away from certain rooms and set clear boundaries. Get them into a good routine each day.
For cats, depending on their personality they may take longer to come out of their shell in their new home. They like to explore every inch of your home for great nap spots, other smells too if you have children or furkids. Setting them up in one room of the house first with their food, litter, cat posts and toys is ideal so they can get used to the surroundings and activities of the home. Then slowly introduce them to other rooms, family members and existing pets. Read more about introducing your cat to other furry family members here.