General Care
If you are going to introduce a new cat or kitten into a household with other pets, you need to do this slowly.  The new cat should be confined to a room so that he can get used to his new surroundings - moving is never easy on a cat.  Blankets or toys with the scent of the other pets in the household should be given to the new cat and vice versa so that they can get used to each other's scent.  Allow them to sniff at each other through a closed door.  When you decide that they can meet face-to-face, do this in small amounts of time until they tolerate each other.  A great way to get them to interact is by playing with them - I especially suggest a feather wand, fishing rod toy, or string.  Throw the toy in between them, get one cat interested and move him toward the other.  This will take work and daily playing, but soon they will be playing together.

Never leave new cats alone in a house with the older residents.  Put them in their own room with the door securely closed to eliminate problems while you are gone.  Make sure to give equal attention to both old and new cats to avoid them competing for your attention.

Feeding
After they are getting along, your cats should be able to share bowls.  If you prefer not to do this, set up two separate feeding areas for each of your cats to avoid one eating the other's food.

Litterbox
If you adopted your new cat from a litter or a shelter, he will already know how to use the litter box.  When he's in seclusion get him his own; when you bring him out into the house, place him in the litter box and let him get out himself.  He will remember where it is and you should not have any problems.

Health
Your new kitten/cat should have a visit to the veterinarian before being brought home to check for any diseases and to get him up-to-date with his shots.  There are some diseases that can be passed between cats so please be sure to take care of this right away so you don't have health problems down the road.