Hi welcome to the forum
I think I can see your problems straight off though as Katty said exact figures of ammonia, nitrite and nitrate are needed to solve the problem.
First thing Im going to say is I would stop listening the the fish shop, while there are a lot of very knowledgeable fish store workers out there there are more that dont really know much about fish and will sell things to anyone and as a result the fish die so you go back and they make an other sale. Its a very cynical way of looking at the ornamental fish trade but it happens all the time and has happened to just about every forum member here.
The reason I say I think that your fish shop falls into the bad category is because they have been selling you goldfish for a year by the sounds of things. Now this is a problem because your tank is 10 gallons, and although most people will say goldfish are fantastic fish for little tanks and bowls its not true. They actually need very very big tanks its usually recomended that for fancy goldfish you have 20 gallons for the first fish then a further 10 gallons for every goldfish you add to that. With common goldfish and comets you want 20-30 gallons for the first fish with 20 gallons for each fish after that ideally common goldfish need to be in ponds really they grow steadily to a foot long and are pretty active.
The thing is that all the tales about goldfish staying small in small tanks is only true to an extent. When they stay small this means that they are stunted which is due to them releasing a hormone into the water that helps them sense their surroundings to a point that they understand how much they should grow. Most fish give off this hormone but some use it stronger than others. Goldfish are also very messy fish and they poo a lot which means that there is a constant ammonia source in the tank that the filter will battle with and in a big tank it would be in fine but in a tank as small as ten gallons its going to be really over powered so I could easily imagine you have some low ammonia levels in the tank which could be killing your fish but also your filter could be coping but as a result the nitrate (the end process in the cycle) could be very very high and again could kill your fish.
I started keeping goldfish as my first fish but I quickly ran into similar problems. And I understand your frustrations and I guess the most obvious question is how do have so many people kept goldfish in tiny bowls for years and not had these problems. I think the main reason is that a few years ago goldfish were a lot hardier than todays fish, goldfish today are bred so much that the genes in the breeding stock are really really low and also the clinical breeding areas are so sterile that the fish born into them dont gain defences as readily as their wild counter parts. While goldfish in the past were able to live in very very bad conditions I really doubt that most of todays stock is able too and then also there is the question for the owner to ask which is, do you want to keep the fish like this in what is essentially an outdate cruel way to keep fish?
So that said, I would probably recommend one of two routes. First one is getting a bigger tank for your fish, if they are fancy goldfish get a 30 gallon with a nice big external filter that will house the 2 remaining fish nicely. Then by using the API test kit monitor the levels of ammonia and nitrite and when they get above .25 ppm do a large water change with water conditioner. This will be whats called a fish in cycle.
Second route to perhaps consider, this is probs the best option if your not able to get a bigger tank. This would be to take back the goldfish and start whats called a fishless cycle, this is where you use pure household ammonia to cycle the tank rather than using fish poo to do it. This means there are no fish present in the cycling stage which means you wont have any risk of loosing fish early on. It also means that when you set up the tank with fish you will be able to do so in good faith that your water will be okay. With this route I would also go tropical with the fish and avoid the goldfish, you said you have a heater and honestly there is no difference between goldfish and tropical fish at all. Also having the tank empty for a few weeks while you do the fishless cycle will mean that if there is a disease or parasite present in the tank it will die off while there are no fish to support it in the tank.
If you did go with route two you would have a lot of choice in the type of fish you could keep and with the help of this forum you will be able to do so very successfully. The best way to long term successful fish tanks is research, learn all about your fish before you buy them then you dont leave your self open to bad advice from fish shop workers and you also gain an insight into the creatures you are keeping. It also means that you understand what fish can be kept together and also how big they get. One of the biggest problems in hobby in a similar way to the goldfish is that some shops still sell baby massive fish to small tanks like yours. The amount of 10 gallon tanks I have seen full of Iridescent sharks, common plecos and bala sharks is shocking and all of those fish grow bigger than a foot long with the shark making 3-6 foot as adults.
It seems like you have had a really bad start to the hobby but now your using the internet you will be able to learn so much that the fish shops wont ever tell you. Hope what I have said hasnt put you off, fishkeeping is a very rewarding hobby when it goes well.
Wills
