Fear is common and a perfectly normal, innate, and adaptive behavior in all animals. However, if fear isn't addressed, it can develop into serious behavioral and health problems. If your cat is showing fear or shyness, deal with it proactively.
Many cats are fearful at first in a new environment and need a little time to adjust. Cats that are fearful of noises usually do best in relatively quiet homes. Many fearful cats slowly become more confident as they get used to the routine noises in their living space. Going to a new, strange environment can cause a fearful cat to regress at first in a new home. The amount of time it takes a cat to settle into a new home varies from case to case. While some cats adjust quickly within a few days, others may take a week or even months, depending on their individual personalities. It helps to first confine your cat to a small, quiet room, ensuring that your cat has appropriate hiding places. Provide hiding places that are easily accessible and comfortable. Place the litter box within easy reach of the cat, but away from food and water. Keep your cat confined until they feel comfortable in the room and shows signs of wanting to explore the surroundings. Let them explore the rest of the house gradually (too much territory all at once may be overwhelming).
If at any point during this process your cat regresses, confine them back to their "safe" room for a few days and start over by only allowing access to one room at a time. When your cat is ready, they will come to you. Encourage play with interactive toys (e.g. cat dancer or fishing pole type toy), but make sure that the toy you are using is not loud or scary. Some cats are very play-motivated and regular play sessions can help bring them out of their shell and out of hiding. Do not startle your cat. If you have to do anything noisy in the house, (e.g. vacuum, moving furniture, having a dinner party), confine your cat to their "safe" room. Keep a white noise machine in the room, play bird sounds or classical music to help calm your cat.
This document created by the San Francisco SPCA with a grant from Maddie's Fund®.