People, you're getting into debates with each other and using abbreviations like LFS, when Lubfitchets said that she's brand new and doesn't know much about the hobby.
Lubfitchets -- If you're still reading this thread, in answer to your question, we can't tell for sure what is making your fish sick. There are too many possibilities. But there is a good chance that the fish is being poisoned by toxins in the water, probably ammonia or nitrites, which are the natural result of keeping fish in an enclosed container. You can and should know how much of these toxins are in your water. The ideal level is zero. Even a little ammonia or nitrites will kill a fish over time, sometimes quickly, sometimes a little longer, but both are deadly. Testing the water is the only way to know how much is in there. Buying your own test kit is a good idea, as some have suggested -- that makes it easier for you to test regularly. Your local fish stores (LFS) will probably sell test kits, and many will test your water for free if you bring them a sample. As was mentioned, it's also good to know the level of nitrates and the pH of your tank water, although these are less likely to be the problem in your case at the moment.
The ammonia and nitrites will drop to zero when the tank is "cycled". There is a lot to learn about keeping fish alive and healthy, but it's not too hard to do once you learn the ropes initially. A good place to look for info is this post:
http://www.aquariacentral.com/forums/showthread.php?t=12297
It is also true that it's not a good idea to keep a goldfish in a tank that small. A male betta would be a nice choice. Once you have stable and good water quality, if you get a tank divider, you could consider adding a second male betta. As long as they can see each other but not actually touch each other, they would be inclined to "flare" at each other, and you would have a colorful display. Two bettas is about the max you could keep in five gallons, if you want them to stay healthy. Some may already say that is overstocked, and I've never kept bettas myself. But if you keep the tank well-maintained, and if you start with healthy fish, I think bettas would probably stay healthy and happy with 2.5 gallons per fish.
Finally, it's obviously important to
start with healthy fish. Your current fish may be dying because it might have been sick already. When you buy new fish, try to buy them from a store where all, or at least most, of the fish look healthy. Check around at different stores. If you switch to bettas, try to avoid buying fish that already look half-dead and have ragged fins. Since bettas are often sold in little cups, you may have to look in several stores before finding some that look good.
Just my two cents. Welcome to the hobby, and welcome to Aquaria Central! :hi: