THE NEW SPACE
With most cats, it’s extremely important to start the cat off in a safe room where no one else goes, so it can start to get familiar with its new space. Of course, it will need its food and water, plus its own catbox.
If the new cat’s food and catbox are from its previous life, you’re a step ahead. 
Also helpful is something comforting from its previous space, like a blanket. See that it’s settled, then leave it alone for a few hours.
Some cats adapt quickly, some not so much, but the goal is to get the cat to where it’s calm, even bored. The cat will tell you when it’s ready for the next step with normal calming and positive behaviors like purring, kneading and grooming. It may ask to be let out, but make sure its fear is really subdued before you go onto step 2.
Also make sure the new cat is completely healthy before you make introductions. Cats can have serious communicable diseases, so you need to protect your home pets as well as protecting an unwell animal.
