When you enter your new household, you will need a very different approach to a resident cat or cats than you use for any dogs. In the absence of other evidence, your keywords should be respect and cooperation.
Here's the most common scenario: You are being brought in to fill a vacancy. Prior to your arrival there were two cats of similar ages. One of them died. The humans believe that the other cat is lonely. You are there to provide the surviving cat with companionship.
The resident cat, however, may have an entirely different perspective.
1. She may not have been fond of the deceased cat at all. Many cats, despite mutual dislike or indifference to each other, recognize the importance of a show of solidarity against other species. Also, since cats are place-oriented, they prefer a peaceful environment and make arrangements to tolerate each other.
The remaining cat, however, may have been looking forward to sole rulership of the household. Your arrival disrupts her plans for domination.
2. Nothing makes a cat feel his age like a lively kitten. It's not your fault that you're young, adorable, and highly energetic, but stop your prancing for a moment and consider that the older cat has the right to feel insulted. He may wonder if the humans were dissatisfied with this older model. He may feel that they've been disloyal in the face of his years of faithful service (or behavior that looks to the humans like service, which is good enough).
If the contemplation of this attitude makes you impatient or irritable, consider this: Someday you will be in this situation, an older cat whose life and status are about to be disrupted by a charming little ball of fur. Put yourself in this position. How do you feel? Not so wonderful.
If you keep the older cat's attitude in mind, you can form a successful and indeed vital alliance. An agreeable older cat can fill you in on information critical to your future. She can, for example, give detailed personality profiles about human residents in the household and update you on the training program currently in effect. Together, you can accomplish more than twice as much.
Therefore, your first priority should be the winning over of the current feline resident.