Summarize:

Do not pull the kitten out of the cat carrier right away; leave it in there with the door open and allow it to exit the carrier at its own pace. Make sure it sees its food and water bowl. This may motivate the kitten to leave its carrier and explore. When the kitten does come out, gently pick it up and place it in the litter box to give it an opportunity to go to the bathroom. If the kitten doesn't go, at least it will know there is a litter box for it use. If you have other pets in the house, keep the kitten in its own room (along with litter box, bed, food and water) for a week before introducing him to the other pets. If you have small children, instruct them to be very gentle when they pick up, hold, or play with the kitten to avoid injury to either the kitten or kid. Just let it take its time and it will get used to you and the house soon. Keep the kitten in one room (unless you are there to watch carefully) and other pets away from the kitten for a week before introducing the cat to the other pets. After a while, try playing with the kitten but don't force it to do anything it doesn't want to do. It should come out soon enough, and don't panic if it comes out of the pet carrier and hides under something. Let the kitten explore the room. It will have a lot of sniffing to do. Kittens are pretty smart and will take to using the litter box in no time. That is, as long as they can get into it, it's cleaned daily (scooped), and it's in a quiet spot.
Place the kitten near its bed. Keep the kitten away from other pets at first. Give your kitten time to adjust to its new home. Place the kitten in the litter box every couple of hours until it uses it.