Fish produce ammonia like we produce urine. They are constantly releasing this into the water. Ammonia is _very_ toxic. Fortunately, there exist a few species of bacteria that "eat" ammonia. These bacteria will grow in your filter, and start to reduce the ammonia level in the tank.
Unfortunately, these bacteria release nitrite as they "eat" the ammonia. However, there are other bacteria that will grow to eat that nitrite, once the level gets high enough.
You have a couple problems right now. One, is that it takes weeks for those bacteria colonies to establish themselves, so in the mean-time, nothing is getting rid of the ammonia.
The second problem is that you have a _ton_ of waste being produced in that tank. A ten gallon tank isn't a suitable home for even 1 goldfish, let alone 13. Goldfish are very messy fish, and they get very big (and live for a long time).
So, the lack of the good bacteria, plus the amount of waste being added to the tank means that your ammonia level is probably sky-high.
Now, what do you need to do... First thing it to do a large water change. Remove 50% of the water, and replace it with fresh, treated, water. Do this again 30 minutes later.
Second, pick up a test kit from your local fish store (LFS) and find out what your ammonia and nitrite levels are.
Finally, I would really recommend you return the goldfish. They aren't really suited to a 10 gallon tank. You'll have a much happier tank if you get some smaller fish like Zebra Danios.
If I were you, I'd take the goldies back and get 2 danios (make sure you change _all_ the water before adding new fish to the tank). Over the next few weeks, monitor the ammonia and nitrite levels and do a water change if they get too high. After the ammonia and nitrite levels are 0, it's safe to add a couple more fish and repeat the cycle process (it will be much shorter the second time, since the bacteria will already be there, they just have to grow a bit).
~JW