Spring and summer are the peak times for bringing a new kitten or cat home. Kittens are generally better accepted by the existing cat or cats but some established cats don't like any newcomers on principle. Patience is key--the transition can take several weeks but planning ahead can reduce the stress, allow for an easier transition, and build a positive relationship between your feline companions.
Isolate your new cat in a separate room with its own food, water, litter box, bedding and toys for the first few days. Bring familiar items from the adoption centre in order to make it smell comforting and "homey" for them. Keep the carrier open so the cat has a place to hide. Isolation allows both cats to get used to the scent and sounds of the other cat without risk of confrontation. Be sure to spend a lot of time with each cat or group of cats.
Once all cats in the home seem relaxed, gradually move the food dishes closer to the door that separates them. If you notice any signs of stress, go back to the step where they were comfortable and work more slowly. You can also use a toy for them to play with under the door when they are calm and curious. If the cats are calm, take a cloth/blanket and wipe one cat and then put that cloth in the room with the other cats. Do the same for new and existing cats, so that they both can smell each other in their own areas. If this is comfortable to all cats, mix the scents on one cloth, wiping first one cat, then the other. Reward all calm behaviors with treats and praise in a soft voice.
When the cats are comfortable with the above, try a brief interaction. Crack the door open an inch so that both cats can see but not touch each other. If one cat hisses or tries to attack, close the door and back up the process, and restart more gradually. Sometimes it can be helpful to distract the cats with food. An eye and hook latch or doorstops on each side of the door might help.
When all is going well, place the new cat inside the carrier and allow the other cat(s) to see and smell the new cat more closely in a safe environment. Continue to reward calm behaviors with treats and praise in a soft voice.
If the cats seem comfortable in this environment the next step is to try placing them in the same room with direct supervision. Start introductions for brief periods making it more likely that the experiences will be positive. Remember to be patient and go back a few steps if necessary and gradually re-introduce.
Even when the cats are successfully introduced, remember that each cat needs their own resources ie food, water, bedding and litter boxes, preferably in different locations.