how to welcome your new cat or kitten
Adoption centres regularly hear that a new adopter took home a cat or kitten and assumed that their current cat (or dog) would instantly welcome a new friend. Instead, one or both instantly react badly and everyone is stressed.
The chances of two pets (especially adults) immediately liking one another are slim.
Have a plan
You will hear that the cats should be separated and while not as easy in, for instance, a small apartment, it is doable. If nothing else, a bathroom can be used as temporary quarters for the new cat.
Your new cat you will need its own litterbox (the rule is one per cat, plus one), separate food bowls a scratching post and toys.
Basically, you separate the cats by putting the new cat in a spare bedroom or other space. The cats will be aware of one another but no direct interaction. After a few days, you let the cats see one another from a distance. If the second step is going well, then the cats can meet face to face. For specifics, read these articles to help you form your own introduction plan. The Hills article has suggestions for introducing a cat to your dog(s).
Petfinder: Simple Rules for Introducing Two Cats