As I am writing this, Mollie has been with us for two weeks and the process has been pretty painless. Our two cats are used to other cats, but Mollie has always been an only cat, which led to her keeping her distance from the other cats for a few days.
She soon came out of her shell and for every day that goes by she gets more and more friendly with our two. Nose bumps, sleeping on the bed together. But she does still run when Ziggy wants to play.
Fluffy is 10 and nothing has ever bothered him so he just wombles along, sleeps where he wants and is not afraid of anything. Ziggy is 4, and had a traumatic background (was rescued from terrible physical abuse) so he is jumpy but playful. And Mollie (2) is an absolute sweetheart – just getting used to other cats, a new home and a new family. But she is doing it like a champ. She adores people and cuddles.
There are different theories about how to introduce new cats to each other. Keeping them in separate rooms for a week or so etc. I have never put a new cat in a different room, but have always made a little nest for the new arrival (dark, quiet, with food, sand and a bed) and then let them all get on with it. The new arrival can go to their safe place while they are adjusting. For Mollie, we had planned my office as her little patch but she chose to nest under our bed. We put pillows, sand, food and water under there for her, and she ventured out when she wanted to. After three days we removed her food and sand and she happily came and ate with the other cats and used one of our two litter boxes.