The Rule of 3s
The rule of 3s refers to the expectations that all new cat owners should have when it comes to helping their kitty settle in to their new home!
The first 3 days are going to be the hardest for your new cat. Set up a small room in your house to be your new cat's temporary room. A bathroom, spare bedroom, or office is perfect. Put everything they may need in that room: a bed, toys, a scratching post, a litter box, food, and water. From this room, they can get used to using their new things, and get used to the smells of the house without being overwhelmed. Spend time with your cat while they're in the room to give them a chance to get to know you as well! If your cat spends the first 3 days hiding, that's normal. Give them the space they need to become comfortable with you and their new surroundings.
By 3 weeks, your cat may start venturing out of their room. More confident and outgoing cats may have been out exploring for weeks at this point, and shyer cats will likely just be getting brave enough to explore. Both are normal; it all depends on your specific cat.
You should be able to move your cat's things into different areas of the house at the 3-week mark. By this point, your cat will be accustomed to the scents of their own things, and recognizing the smells in other areas of the home will help them become more comfortable. This is the phase when your cat's true personality will start to peek through any residual shyness!
By this point, your cat will likely be fully accustomed to your home and your routine. This is the stage when you can expect your cat to start waking you up if you sleep in past breakfast time!
It's important to note that being comfortable in the space does not necessarily mean comfortable with the other people and animals living there. This is the step when most cats will start to build bonds and attachments. Building attachments with cats is a long process that requires a great deal of patience, but is definitely worth it at the end when you have a friend for life!