This is all based on my experience and may not apply to everyone. Every cat is unique and may have a difference experience than mine.
Getting a new cat can be exciting and it can be stressful, you never know what’s going to happen with them. They could be very excited to be with you or they could be scared of being in a new place. My first suggested is to try very hard to not scare them and help them get to know their surroundings. It’s best not to just let them be and figure it out on their own. They are in an unfamiliar environment and will likely need help finding the food and water.
It can either take a long time or a short time for cats to get familiar. You may notice that they will slowly start to explore their area while also hiding, do not do anything that might scare them. If they are taking a lot longer than expected to get familiar and used to the environment. It can be helpful to bring them around the house and introduce them more. Be as gentle and calm as you can be when doing this as they are already stressed about being in this new home. You don’t want to provoke them and create more problems than there already are.
If you have multiple cats you might find that they fight at least once or twice and in my experience its inevitable. They smack around once and then they are fine, although if it gets too serious you should try to stop them. If they constantly fight maybe consult someone else that’s not me. There could also be some cases where nothing happens and they just get along.
Always remember to keep the food and water dishes constantly filled. Also put the dishes in a place that’s easily accessible. If you have multiple cats make sure put down more bowls so they don’t feel any pressure. It’s possibly best to have a bowl for each cat, try to make sure to keep them all filled at all times. It may take a while for them to realize that they will always have food, so just make sure that they get comfortable around the house, with others, and with the location of the food/water dishes.
Never move the food dish, water dish, or litter boxes. Cat’s are a routine animal and they do not like change. Moving these could confuse them and change their habits and schedules. Moving the litter box could also result in them being confused and then going to the bathroom on the floor, they would also be very confused so unless it is necessary do not move it. If it is necessary, keep a watch for them and try redirecting them to the new location when they try to go to the old location.
Never bathe your cats unless completely necessary, they do not like water and will retaliate when you try to put them in. They are already perfect at keeping themselves clean and bathing them may lead to more mistrust. If you do decide to bath them, do not use shampoo specified for humans, it can contain toxic material to them. Also do not use too much water, they are already overwhelmed and it might increase their stress level.
Please note that I am not a professional and some of this information MAY be incorrect. I am just an animal lover sharing my experiences and notes.