Okay, so, it looks like you've been given plenty of good advice. When you say common goldfish, do you mean the squat ones with fancy fins or one of the longer, thinner ones with shorter fins? The squat ones can manage in a 30 gal tank (not forever, mind you) and would prefer a 55 gal or larger (gal = gallon). They are also more able to adapt to warmer water then the "true common" goldfish (longer and thiner). True Common Goldfish need ponds or massive tank, and are coldwater fish through and through. They need 100+ gallon tanks, massive filtration, and colder temps then any of the other fish in your tank. The rest of your fish are incompatable with True Common Golfish.
What I recomend you do is get a larger tank, between 30 and 55 gallons, as well as 2 Marineland Penguin 350's. The goldfish goes into the larger tank by itself (after the cycle has completed) and the tropical fish can stay in the 10gal. The larger tank gets both Marineland Penguin 350's, to keep up with the goldfish's massiving waste.
The proccess of cycling is rather simple. There are 3 types of waste produced over time in an aquarium. Ammonia is produced diractly as fish waste in the form of urine (pee) and feces (poo). Waste fish food (whatever isn't eatn) also decays and produces ammonia. The first stage of cycling is where the ammonia eating bacteria grow in numbers. They turn the ammonia into nitrites (I have dyslexia, so someone correct me if it's really Nitrates) which again build uop and are harmfull to fish. Then the nitrite eating bacteria develope, and turn it into nitrates. Nitrates aren't as bad as either ammonia or nitrites, but should not get above 20ppm on the mast test kit.
A side note: you can use "filter floss", or polyester fiber from your local crafts store as a filter medium. With two filters in a tank, you can replace part of one without fear of causing a massive cycle. I have 5 filters on 1 tank (Yea, I'm a little overboard, but it's an overstocked tank, best be safe) and I rarely change out he medium. So the more the merrier, so long as you don't have fish being blown around by the water flow from the filters.
Water Changes are your best friend right now. Go out and buy another grav-vac or siphion, as well as an extation hose, and then either do it the way they show in the vid or do it my way and just suck on the end for a moment. Make sure to have eithe a drain or a bucket, and always have the end of the hose below the tank bottom. Don't walk away, it WILL drain your entire tank if you leave it too long (happened to me once, thank god for driftwood or I would have had an empty tank).
I'm 17 and have 5 tanks up and running. 2 of those are salt water tank, 3 freshwater. My largest is 40 gallons and need 3 20 gallon Water Changes each week (overstocked). The smallest is 10 gallons and get 1 WC every month (heavily planted, very lighting stocked). It's not as hard as it may sound. Once set up and stable, things rarely go wrong with a tank. You definitly care about the fish, and you want whats best for them. Now, I hope I didn't confuse you at all, I was pretty random in the way I posted that, sorry
Good Luck with your tank, keep us updated, and I hope all your fish make it!
~Siv