Your tank likely has cycled. (but this is not guaranteed. Mine took months to cycle)
http://www.fishlore.com/NitrogenCycle.htm
Here is a good explanation of the cycle we're talking about. In short, it's the growth of the bacteria that eat ammonia and produce nitrites, then the growth of the bacteria that eat nitrites and produce nitrates (which are much less toxic to fish than ammonia or nitrites).
Algae is not necessary. In fact, it can be harmful to fish if there is too much of it. I'm keeping something that eats algae in each of my tanks to keep the population down.
Also, when you find a new home for your goldfish, you can make their transition, and the cycle of their tank, easier by transferring part of the first aquarium's filter to the goldfish's tank filter. (Just taking a little bit of the carbon out of the filter bag, if you're using the kind that rests on the back of the tank, and putting it in the new filter's bag will seed the bacteria, giving them a jump start on growth, if that makes sense.)