Hi Lizzie...here are some simple kitten/cat care guidelines to supplement everyone elses wonderful answers already offered:
1) Have fresh water easily accessible at all times changed daily. No milk as it causes bowel distresses because cats are lactose intolerant. http://www.cat-world.com.au/CatMilk.htm
2) Initially, kittens needs to be medically evaluated and vaccinated for feline diseases and dewormed for intestinal parasites which all cats are susceptible to getting. The following are the vaccinations:
Feline Panleukopenia Virus
Chlamydia, Feline Infectious Peritonitis, and Ringworm
Feline Calicivirus/Herpesvirus
Feline Leukemia Virus
More on feline vaccinations: http://www.sniksnak.com/cathealth/vaccinations.html
More on feline deworming: http://www.marvistavet.com/html/body_roundworms_in_cats___kittens.html
http://www.marvistavet.com/html/body_tapeworm.html
Vaccinations are done yearly, however some vets will say only the first three years are necessary and every so many years thereafter. Costs vary depending on where you reside and the individual clinics.
2a) Cats benefit from spay/neuter just before they reach the age of sexual maturity (e.g. 5-7 months). Problems with unspayed/unneutered cats is they will urine spray, have a tendency to want to bolt and roam, get into cat fights, become pregnant, etc. Spay/neuter is relatively inexpensive (fees vary again depending on where you reside...usually it can be anywhere from $15-75, depending on gender as well) if done at a shelter. If you are located in the USA you can contact 1-800 SPAY USA http://spayusa.org to locate a nearby participating shelter to perform the procedure.
More on neutering: http://www.marvistavet.com/html/body_neutering_the_male_cat.html
More on spaying: http://www.marvistavet.com/html/the_feline_spay.html
3) Kittens benefit from a diet strictly designed for them. Please absolutely no human foods as they contain ingredients which can be toxic to cats. Please see the following "sample" list of foods to avoid feeding a cat: http://www.peteducation.com:80/article.cfm?cls=0&cat=1276&articleid=1029
Dangers of feeding table scraps to cats: http://www.peteducation.com:80/article.cfm?cls=1&cat=1397&articleid=833
4) Kittens are very fragile so if there's any dramatic changes with their appetite, thirst, bathroom habits and general behaviour it truly requires a visit to the veterinary hospital. Diarrhoea in young kittens is the most dangerous as they can become seriously dehydrated quickly and die as a result. Intestinal parasites as well as sudden changes in diet can be the leading contributors to intestinal distress. When feeding cat food it's best to not suddenly change from one type of food to another rather gradually over time. Here's a chart on how to gradually make the switch: http://www.drsfostersmith.com/pic/article.cfm?dept_id=0&aid=100
5) Cats ideally should be fed on a schedule rather than free fed to develop good eating habits as some cats/kittens are known to gorge their foods and overeat. Later in life this develops into cats who become overweight and is the leading cause for feline diabetes.
http://www.vet.cornell.edu/fhc/news/feed.htm
http://www.vet.cornell.edu/fhc/brochures/feedcats.html
6) It's important to kitten proof your home very well. Cats sometimes accidentally ingest things that can be fatal such as string-like objects or small toys. Supervision of string-like toys is highly recommended. Christmas tinsel, for example can be very deadly because when ingested it will twist off the intestinal tract and be fatal. For more on string and other household dangers to cats: http://www.cat-world.com.au/cat-worlddangers.htm
7) Cats live a safer and healthier life as an indoor pet rather than outdoors which leaves them open to moving cars, wild animals, dogs, cat fights with unvaccinated and diseased cats, mean people who don't like cats, etc. More on the dangers of cats who live outdoors:
http://www.cathelp-online.com/health/indooronly.php
http://www.runway.net/b/moonmaid/in-or-out.html
http://www.petplace.com/cats/outdoor-dangers/page1.aspx
8) Cats use a litter box to urinate and defecate. Ideally, a new cat should be kept in a small quiet room nearby their litter box for a few days (younger kittens a little longer as their bladders and bowels are small and hard for them to hold longer so accidents are likely to occur. Keeping the kitten/cat isolated in this room helps them to learn where the box is before venturing on out into the house. This also helps the kitten/cat learn that this room is always safe and quiet to do their business.
There are many types of litter available. Usually most people prefer Arm & Hammer scoopable. However, there are many environmentally friendly litters (e.g. Swheat Scoop, Yesterday News, Feline Pine, etc) now available which can be safer if the cat accidentally ingest when they groom themselves as they do not clump and solidify inside a cat's intestinal tract. It's best to always scoop daily to keep odours to a minimum. Some cats will refuse to use a soiled box so this is important to keep the box cleaned (scooped) daily. Location of the litter box is important...usually a private space is ideal. More on litter box tips: http://www.hsus.org/pets/pet_care/our_pets_for_life_program/cat_behavior_tip_sheets/preventing_litter_box_problems.html
9) Kittens love to play. They love cardboard boxes and paper bags to hide in and attack. Most pet stores have kitten-safe toys and each cat has their personal favourites so you have to experiment which toy your kitten will fancy. Scratching posts are a must as all cats love to claw. The more the merrier. To encourage your kitten to claw the posts play with a toy nearby them and reward them with a food treat. Anytime your kittens claws inappropriately elsewhere gently pick them up and place them next to their scratching posts rather than scold them. It helps them learn that the scratching posts/toys are more fun to do this and learn that anywhere else is not gaining them any attention whatsoever.
Instructional video on how to trim a cat's nails: http://www.felinevideos.vet.cornell.edu/trimming_claws/full_movie.shtml
10) Cats are wonderful in grooming themselves and don't require bathing unlike dogs. Cats have small bristle-like hairs known as papillae on their tongue which distributes keratin a deodorant like substance which allows them to thoroughly clean their entire bodies. It's only when they get into something that is toxic to if there were to self groom that bathing would be then necessary. If you would like to supplement their grooming you can use babywipes.
Finally, here are a couple wonderful and helpful websites on how to best care for kittens and cats:
http://www.kittencare.com/
http://www.catchow.com/KittenCareCenterHome.aspx
Other helpful websites: http://placervillevet.com/kitten_care.htm