EDIT: Looks like Lupin got here first! Well, I second his remarks and add:
Hey Jack,
Welcome to AC! :welcome:
We're glad to have you. I'm sad to hear that your original goldfish didn't make it. It sounds like you started out the way most of us start out - with too many fish in too small of a tank.
Goldfish are very hardy fish, and extremely easy to care for. They have simple needs, but like all aquarium fish, there are some very critical factors that can make or break a healthy tank.
First, goldfish need lots of room. This is the most important part. Fancy goldfish can grow 8-12 inches long and need at least a 20 gallon tank for one, plus 10 gallons more for each additional fish. Common or comet goldfish (the single-tailed types) require even more space, with a 55 gallon for one and 10-15 gallons for each additional fish. Although most goldfish in pet stores are no larger than 3 or 4 inches, common and comet goldfish can reach 18-20 inches long. Many people new to goldfish think that they only grow to the size of their tank. Actually, a goldfish kept in a tiny tank will produce hormones that will slow its own growth, plus the buildup of toxic chemicals in the small tank will sicken it and can cause stunting. This condition causes organ failure, chronic illness and dramatically shortens lifespan.
It sounds like your fish suffered from a tank that was just too small. Small tanks in general are much harder to take care of than larger tanks, simply because it's harder to balance a small ecosystem. With more water, you have more room for error, and small problems or mistakes aren't a disaster.
Once you have the right size tank for goldfish, they need a powerful filter. Goldfish have a pretty crude digestive system. They have no true stomach - the intestine just behind their mouths can expand to hold food sort of like a stomach, but they don't have the acids to break down their food and fully digest it. So they produce a lot of waste. It's important to filter the tank properly and perform regular water changes to remove the waste chemicals. Many of us use the rule of thumb - for goldfish, use a filter rated for a tank twice as large as the one you have.
If you have the option to start with a big tank, you're much more likely to be successful with goldfish. They're social creatures, so two to four fancies in a 50 gallon, or two to four comets in a 80 or 100 gallon would be ideal setups. Large tanks are easy to find on our sister site, Monsterfishkeepers, in the classifieds section. You can also try craigslist or your local classifieds. Tanks big enough for goldfish can be expensive if you buy them new, but gently used tanks are very reasonable.
As for your 10 gallon tank, shrimp, snails or small fish like guppies would do very well in that setup. If you have used copper-based chemicals on the tank in the past, snails or shrimp will be poisoned by the residue. But a small group of fish would be very happy. Beyond that, you should not need to disinfect the tank at all. Just start with new water and clean the filter. Be sure to add new carbon to remove any residual medications or chemicals, and you're set.
Best of luck! Let us know if you have any questions. We're here to help.