Thanks everyone very much for the responses.
Here's what's happened since then: I went to a nearby independent pet shop in the few minutes of spare time I had. The man came to help me and I explained that I was given a goldfish in a 1 gallon tank, and I asked if it would be alright. He said I'd just have to change the water frequently. I asked how to do it, he said put the fish in a bowl and dump out the water. I asked about a siphon, he said there was no need. I asked about test kits and he said there was no need, they would just help me see the cycling happen. He sold me a net and de-chlorinator (I already had that but oh well) and I left. He didn't seem concerned at all about the size of the tank. I'm assuming he is just apathetic, because EVERYONE here has said this tank is much much too small.
I just did about a 67% water change in the tank. I took him out with a net, put him in a bowl that had been sitting with water for a good 4-5 hours, and dumped the water in the sink.
This tank is a set up where there is a pump underneath the tank's podium. It makes bubbles via a hose going up the back of the tank and onto a fan-shaped opening. This fan shaped opening is held up in the tank by a hollow plastic "column." The column attaches to the "floor" surface which is slotted plastic. This entire assembly came out, and because gravel had gathered around and under it, it is now not sitting very well.
After the water change, I can still see lots of poop floating around. I fed him one flake of food and I don't think he ate it; I'm sure he's stressed out.
He has been frantically swimming around the tank looking as though he lost something. I just came to check on him again and to take this picture, and he has settled down a bit. I do think the camera flash scared him again.

I feel terrible doing this to the poor fish. It will need another water change probably tomorrow. Should I just buy a bigger tank? If so, what size exactly? The pet shop I was at had a 15 gallon "starter kit" with some various decorations for $45. I don't have tons of money but I'll be damned if I just let this thing eat it's own **** and die. I didn't ask for it but now that I have it I'm not going to be cruel.
Here is a picture of Hank in the tank. Let's play "how many things can you find wrong?" That way I can learn and have ammo against the parents when they see I've purchased a new tank.