If you keep the QT filter running in the main tank, you won't need to carry out a fishless cycle, no. However, to save a little on your electricity bill, you could also just keep the media (the stuff inside the filter) in the main filter and take some out when you're ready to get your second lot of fish.
You're on the right track BUT . . . If you're using the fish food method, the best way to do so is to get a pop sock and add a handful of flakes to it. Tie it in the water near the outflow of the filter/s and leave it there til ammonia reads 2ppm, then remove, allow ammonia to drop a little, and basically repeat with a fresh ammonia ball.
Alternatively, you could try to get bottled ammonia from the cleaning aisle in the likes of Homebase or Boots (IIRC).
The cycle will take forever if you're just adding a bit of food everyday
I say common goldfish - I actually don't know what they are but they were bought from a reputable pet shop who has been in the area since I was a child (34 yrs ago) (according to pets at home my results are) Ammonia: 0 Nitrite: 0 Nitrate: 10-20 ppm pH: 7.6.
600L/H Aquarium Internal Fish Tank Filter Pump All Pond Solutions 600IF - thought it would be ok got it off Amazon.
Partial water change done every 2-3 days varying between 20/30% and then a 50% once a week and full change every two- three. Always use a dechlorinator - gold tap safe.
Tank was'nt cycled prior to getting these two but they went into the same water they came in as I asked for extra bags of their water from the shop so it wasn't a shock to their system.
Had planned on getting a 20 gallon (ish) tank for the two of them however if they can't be together I will get another 10gallon one for the gold one if it survives.
The pet shop said that an additional oxygen tablet wouldn't hurt the fish
You say the pet shop told you the readings were ammonia and nitrite 0, nitrate 10-20ppm. Did they actually write the results down for you, or did they just tell you it's fine? Ideally, it's best if you can have your own test kit, such as API master test kit.
Carrying out a full water change is unnecessary and can shock a fish, which in turn, can prove fatal. The most you'd need to change, realistically, is 50% and even then, I'd say that should only ever be in emergencies.
10gal is far too small for a common goldy They grow to around 1ft in length and really are pond fish. Even fancies should be in a 30gal tank (and that's for 1). Sorry, I know it's probably not what you want to hear.
The Prime will be protecting the fish from the toxins, but obviously, as you know yourself, we'll need to find out why.
Could you post the latest results? Any new additions to the tank recently? If so, what are they? How long has the tank been set up? (Sorry, you've probably gone through this already). When was the last water change? Did you use the Prime then, or some other dechloronator.
First of all, try not to panic. He'll be fine with only 1 eye. A mild antibacterial such as Melafix or a Mela/Pimafix combo should protect the eye and keep infection at bay until the wound closes over.
Apart from that, basic tank mainenance is now even more of a priority - ammonia and nitrite need to be kept at 0ppm, nitrate as low as possible (I know, that won't be easy with 3 goldies in a 60ltr) but definately under 40ppm.