It seems that there are some people out there who have experienced sick fish while trying to keep them. I would guess that most people have, especially when first getting started. I will address some common mistakes that can lead to sick fish when you first set up your aquarium.
The first is cycling the tank. When you put your fish in your aquarium they will put their waste into the water and they may not eat all the food that you put in to feed them. This waste and excess food breaks down to create ammonia. Ammonia is toxic to fish, you don't need very much to make the fish sick or kill them. There are natural bacteria that will eat this ammonia, but it takes time, about 2 weeks, for them to grow, so in the mean time you fish are being poisoned by the water. These bacteria excrete nitrites as waste which is also poisonous to fish, but no need to fear another type of bacteria will come that eats the nitrites, again about 2 weeks, this bacteria excretes nitrates. Nitrates are only harmful to the fish in large amounts, the amount varies by type of fish so find out how much is okay before bringing the fish home.
So if you get an aquarium bring it home put water in it and your fish, you will be poisoning them for about a month, which many fish can't survive. Some fish that can survive the poisoning are weakened and get sick with illnesses such as the infamous ich and die anyway. To prevent this you should cycle your tank before introducing your fish.
To cycle your tank what you need is your tank, some pure ammonia, fish food, a shrimp, or some other seafood meat uncooked, and a water test kit. The water test kit is very important so if you don't have one you should go to you favorite fish store and get one. Once you have your tank all setup how want it, ie got your substrate, plants, other decorations, filter and heater in place, etc. and all filled up with water you add the ammonia or ammonia creation stuff. The process will be faster if you get pure ammonia, don't use the stuff with perfume added, because the fish may not like the perfume. Add about 1tbs of ammonia to the water (depending on the size of tank, check aquahobby in my links section for more specific details, they have very high quality resources there) each test, and test the water about 1hr or so after. The numbers for ammonia should go up then down, and when the ammonia starts going down the nitites should start going up. When they both get to zero do a partial water change to get rid of the nitrates and your tank is ready to add fish, but don't add to many at a time.
You should do partial water changes when the nitrates get to high (20-40 ppm depending on type of fish), you shouldn't have to worry about ammonia or nitrites but do continue to check them.
If you use an ammonia maker, instead of just ammonia it will take longer to get ammonia into the tank because the ammonia maker has to break down. Once the cycle is done remove remaining ammonia maker from the tank.
Stress is anothing thing that contributes to fish getting sick. Some things that contribute to fish stress are temperature swings, change in water pH, being moved, or any other big change in their environment. So try to keep the water the same by using a good heater and make sure the water you add to the tank is the same temperature as the water already in the tank. Try not to use pH adjusters in your tank, instead get fish that like the pH of your water.
One thing to keep in mind when adding water to your tank is to use a dechlorinator, the clorine in tap water will kill the bacteria that is keeping the toxins out of the water and can damage your fish's slime coat.
