Welcome to AC
Great job upgrading to the bigger tank. There is now absolutely no reason why you cannot sort this out.
I would say the first tank was not cycled anyway so you were probably headed for problems anyway but major, major kudos for getting the bigger set-up.
At this stage it probably seems like you are at the bottom of a very steep learning curve but stick with it it is actually pretty straightforward, and just requires a bit of effort - you clearly have the commitment required to make a great go of this.
First things first, the ich, and yes it does sound like that - can you post a pic ?
If you are confident that ich is what you have (look at pictures on the internet) do a site search here for treatment - heat and salt is your best option.
Next, your tank is almost certainly cycling and you need to manage that too. Can you confirm what kind of test kit you are using ? If its not a good liquid drop kit its pretty useless and you need to buy the API Master Test Kit - the single best piece of kit you could ever buy at this stage.
Your readings are that of a cycled tank
if you are happy that you have a good liquid drop kit
and that the test is being carried out properly
and that your tap water is 0 for nitrates - test that now and come back to us.
On cycling, in short, your fish are producing ammonia, which is extremely toxic to them. Given some time bacteria will develop to transform this to nitrite, also extremely toxic. Given further time, more bacteria will transform this to nitrate, toxic but only in far higher doses.
What you need to be doing is a large water change every time you have ammonia or nitrite at levels of .25ppm or more. You won't know that this is so unless you have a good liquid drop test kit. It might take a couple of large water change (50% or more) per day.
This is going to be very tough to manage with the ich treatment but you have no real option.
I know your test results are showing 0,0 and 10 which doesn't indicate a cycle taking place - if you can confirm your test kit and what your tap nitrates are it might be that you don't have this issue to deal with.
If you didn't have ich in the tank I would be recommending that you bring back the new fish to where-ever you got them and try and manage just the one feller you have left from your initial present through a cycle (presuming that the tank is actually cycling) but that is not really an option with ich in the tank.
As an aside what you never do with an uncycled tank or if you suspect that it is cycling, is add more fish - you don't have sufficient bacteria in the tank to cope with the waste currently being produced, so the last thing you do is add more. Just mark that one down to experience, as you are where you are.
If you can post a few pics of your fish we can also work on identifying them.
What filter and heater are you running ? If the filter came with the 10G you will probably need to upgrade that, and with heaters I always go on the basis that two is one and one is none, i.e. a bit of built in redundancy helps.
And welcome to AC - you found a good site