Hi nufishlover
I'm not sure how long you've had fish and how much you understand about the cycling process and the role the beneficial bacteria, so please do not be offended if I oversimplify things.
Let's start with the basics. If you read this article it will give you some idea of how cycling works and what the bacteria does in a tank:
http/fish.orbust.net/forums/index.php?sh...showtopic=10099
Do you have a set of test kits for ammonia, nitrite and nitrate? If not, it would be a good idea to buy them so you can monitor the progress of the cycling.
The bacteria that caused your fish to become ill were most likely Columnaris and/or Aeromonas. These bad bacteria will always be present and the best way to prevent illness is to keep them to a minimum by frequent water changes and gravel cleanings. They will not attack strong and healthy fish, but if your fish become stressed by overcrowding, high nitrite content in the water, uneaten food left to spoil, sudden temperature changes, aggression by other fish, or infrequent water changes, they will take advantage of the opportunity. In other words, don't worry about them if you follow the rules for good fishkeeping.
Since you will be setting up another tank, and keeping in mind that you want corys, and corys require a fine gravel rather than pebbles or such, this might influence whether or not you will want to move the gravel you have in your 10 gallon into your new tank. What kind is it?
These things would be helpful to know before I go further. If your gravel is appropriate, and you can move it into the new tank, you will have a great advantage in cycling the new tank..
I think I read that you just cleaned your filter and changed the media. It would have been better if you hadn't, but either way, keep it going and new bacteria will colonize it. If your new filter does not churn the water too much, get that one going in there too. Put the plants, decorations etc. back into the tank as well, because they will also become coated with beneficial bacteria and the more you can move to the new one, the easier cycling will be.
If you post back, on this thread, I'll be happy to help you all I can.