TheKoiMaiden is doing a bang up job helping you out, but just to add in some points...
You can speed up/help the cycle by doing a couple additional things here. If you know anyone that has an establish, HEALTHY tank, you can borrow some "bioseed" to add to your tank. "bioseed" can mean anything on which beneficial bacteria can grow (filter media, substrate, decorations, plants). This addition can seed your tank with bacterial and cut down on the time it takes you to multiple your own bacteria. Just make sure you only use seed from a healthy tank, no diseases or illness (past or present)!
Another thing to help is to add plants. With goldies being the little plant eating machines they are, this can either work or just ending being a snack. So choose plants carefully! I would reccommend two plants- duckweed and hornwort. (You can click on the shaded names to read more.) Both are fast growing, so they suck up (at least some of) the ammonia before it gets a chance to hurt your fish. Duckweed is a surface plant, and goldies do love to eat it. Its nutritious! So just make sure you get a large enough portion that they fish don't eat it all up before it gets a chance to start to multiple. Hornwort is a floating plant, but it can also be anchored. It does shed its needles annoyingly under lower light, so its a bit messy and you really have to keep up on vacuuming up the shed leaves (you should be vacuuming at least weekly, anyway). But even considering the shedding, hornwort is awesome because it grows quickly, can survive low light, and isn't usually eaten by goldies (unless they're desperate, and even then it usually grows quickly enough to survive the damage).
During cycling, it helps to feed sparingly. The less you feed, the less waste is being produced, so the less ammonia (both from fish waste and food waste) is being released into the water. Lower, but still in the comfort range, temps can help too by slowing metabolism.
I think you said you have gravel, right? This should be vacuumed at least weekly. Do you have a gravel vac? If not, one can be made for common materials. There's a tutorial on youtube.com for a gravel vac made from a plastic water bottle.
A great alternative to the 10g temp aquarium would be using a large plastic storage bin as a temp tank. You can treat it just like an aquarium, using your filter and substrate, etc. This will allow you to have a much larger temp home for CHEAP or FREE while you're saving your money for their permanent home. Its not pretty, but its better for the fish.
Good luck! And thanks for trying to look out for fish!
"My dither fish need dither fish!"