The rule in general is 1" of fish per 1' tank space for tropical fresh-water fish. You can fudge that a little one you know what you're doing (different species produce different amount of waste).
Mrr you shouldn't be having so much problem 'cycling' your tank.http://www.school-of-tropical-fish.com/nitrogen-cycle-diagrams.html
You can see here how your 3 main components rise and fall as you start the cycle on the tank.
(You DEFINITELY need to know all about the nitrogen cycle before keeping fish, it will make things a MILLION times easier)
When starting a brand-new tank the problem is that it is sterile, there are no good bacteria in there to eat all the waste the fish produce. The BEST way to cycle a new tank is by purchase a little "used" gravel from the pet store or a friend's well established tank - this is the same thing you're getting when you buy a commercial cycler product.
The #1 sign that your tank/bacteria balance is off is when you see bubbles forming on the surface (from your air-pump) and they are able to travel from one side of the tank to the other with out popping. That's BIG trouble and you'd better do a 20% water change RIGHT away.
A fish tank should also never have a 'fishy' smell, it should smell like a fast-running, clean stream.
If you see the fish looking like they are visible distress (gills flapping, gasping for air) you've also got a problem.
This does not need to be VERY stressful on the fish as one of the articles linked-too suggested. If you change the water out of the system fairly frequently over the first few weeks (you can not do it every day, the bacteria won't be able to get a toe-hold), and monitor your chemical levels everyone should be fine.
Goldfish are very heavy feeders/poop-machines. I'd not really familiar with what you're attempting but I would suggest rosy minnows (also super cheap) instead (don't get as big).
I'm assuming we're using the fish to produce fertilizer for the vegetables?
10c Goldfish are frequently very genetically weak & can be ill to begin with, depending on whether or not you're getting them from a reputable business (a.k.a. NOT walmart). Feeder goldfish can also get as large at koi & VERY quickly.
Obviously you're not going to be in a situation where you want to add medication to the water, or salt, to help the fish out when they do become sick. So it may be best to keep a quarantine tank so you don't loose all your stock.
Fish are just has happy in plastic rubber-maids as they are in glass aquariums, they are just harder to see. When you start getting up in the higher gallon range it's better to go with a more sturdy plastic or the sides will bow!
Always check out your local aquarium club before heading out to the store to buy new! People are constantly giving away (or selling for very cheap) old set-ups that they've grown out of, have too much of, getting out of fish-keeping. A good price for a tank is $1/gallon.