Did you do anything with the tank for the first week, or just let the water run? Unfortunately, it sounds like you got some really bad advice. Just letting the filters run doesn't actually cycle the tank. The cycling process really refers to building healthy bacteria colonies to process the ammonia and nitrites into less toxic nitrates, which are removed by water changes. There are a couple of good articles on this site, ill see if I can find the link.
Are you planning on keeping these fish after the cycle, or did you jut get them based on his recommendation? Goldfish are dirty, get too big for a 10 gallon, and have different temperature requirements than tropicals, so they really shouldn't be kept together. And some people will tell you danios are too active for a 10 gallon, and need a bigger tank. If you're interested in doing a fish less cycle, you may want to return them and just use pure ammonia, or one of the other methods.
If these fish fit into your final stocking plan (see cautions above) you'll have to do daily testing and water changes to keep your water parameters at safe levels. I recommend Seachem Prime, it detoxifies the ammonia so it won't harm your fish, but will still be available for the bacteria to feed on. You'll still have to do frequent water changes, but it'll be less harmful to the fish in the meantime. The cycling process can take weeks, or even months. You also want to make sure you're adding a dechlorinator to your tap water when you do water changes. (Prime will work for this.) Otherwise the chlorine in the water will both harm the fish, and kill off any good bacteria you're trying to grow.
I didn't know about cycling when I set up my tank, and it resulted in many dead fish for me, and months of battling unhealthy water conditions. If I was to do it all over again, I'd do a fishless cycle.
It all sounds overwhelming, but it's really not so bad when you get an understanding of the biology of the tank. Do some reading about cycling, and really consider going the fishless route.
One last thing, check out aqadvisor.com as a guide for stocking. 10 gallons is really smaller than you'd think, so you're a little limited with what you can get. But that doesn't mean you can't still have an interesting tank, you just have to be a bit more careful with your planning.
It'll get better!
