We adopted a traumatized 3-year-old cat last year. When adopting, we were told about the rule of 3.
3 days - the cat basically sticks to one room as it gets acclimated to the smells and sounds of the new home. Our cat was very jumpy and anxious
3 weeks- the cats starts figuring out the family schedule (including mealtimes) and exploring the house. Start introducing scratch pads, toys, bedding. Our cat started loving very active playtime
3 months - the cat has figured out that this is “home” and you are it’s people.