This article is from the Getting A Cat FAQ, posted to rec.pets.cats newsgroup. Maintained by Cindy Tittle Moore with numerous contributions by others.
Generally, a very young cat doesn't need the full run of an entire house. Use your judgement, but leaving it in one room until it is a little older can save both of you some anxiety. A kitten will need a different diet than an adult; most brands of cat food also come in "kitten food" versions. Kittens have small stomachs and big appetites; they need to be fed several times a day.
Most kittens will understand how to use the litter box. Usually their mother teaches them, but they will pick it up easily on their own. If you have a too-young cat, you can teach it by confining it to one room so that access to the litter box is easy and putting it in the litter box after feeding.
You might wind up with kittens too young to have been separated from their mother for whatever reason. If you have an orphan kitten, you will need to provide a warm draft-free area and use something like KMR (kitten milk replacer) for food, using an eyedropper. Consult your vet for advice and help.
From kittenhood, accustom your cat to being handled. Look into its ears (clean, white and light pink), eyes (clear, no runniness, inner eyelids may blink but should remain open), nose (clean and pink (or its normal color) and mouth (clean, light pink gums) regularly. Hold it still and look at its anus; pick up its paws and look at the pads and claws. This will have the added benefit that you will notice any changes from normal quickly and be able to call up your vet if something is wrong.
Do arrange for the kitten to meet plenty of people; this will socialize your cat and it will not hide from people when adult.