One of the first things you'll realise when you bring back a cat, is their curiosity to explore and get as much sensory experience of the new environment as possible.
Although you will have the urge to want to play with your new family member, and even hold it, remember to do your best to refrain from this at the beginning and give the kitty it's own space and time to explore. If it wants to scurry under a kitchen cupboard, or under the bed out of sight, then so be it.
Eventually it will emerge and continue its exploration. Allowing your cat to do this at leisure does not mean that you can't supervise as often the new environment may bring some inherent dangers which your cat cannot get themselves out of. For example, getting locked in a cupboard, or stuck on a high shelf. Remember to handle the cat calmly and gently without making a big scene out of things. Over enthusiastic handling at this point can be very disorientating and may bring out defence mechanisms. Being bitten, or even scratched does not solicit positive feelings about bringing home a new pet, but from the cat's point of view, it was probably justified.
By all means, stroke the cat as it passes by, even talk to it. The reassurance of the human voice will help break down walls, and bridge the gap between old and new homes.
A new cat when tired will attempt to find it's own bed. Let it. Allow your cat to rest undisturbed, and like a baby of your own, make every effort to not wake it or make it difficult for them to get some quiet sleep.