2 gal without a filter is not suitable for goldfish. The usual recommendation is 20 gal minimum .... you might be able to manage one or 2 fancy in a 10 when small, but you will need constant checking & fiddling with the tank to keep that stable.
Do you have a test kit? It is impossible to manage a small tank with goldfish without testing. <And I would recommend a liquid test kit, likt the API.>
When you move him to the 3 gal, did you move everything? substrate, decor, etc.
A tank need to "cycle" before suitable for fish. <Cycle means the establishment of bacterial colony - aka the biofilter - to handle fish waste. Read up on it in the articles section if you are not familiar.>
Without the bacteria, the tank accumulates ammonia, and that is toxic. Ammonia burns the fish, that likely accounts for the black marks you saw. If the ammonia burns gets infected, they can turn white. Possible, that is what your white spots are, or perhaps you have ick or some other parasite brought on by the stress.
The red spots on the pic don't look like costia to me, although this might be an early case that had not spread to the typical areas. I would think that some form of bacterial infection is more likely.
However, I think a lot of the problems are brought on because of water quality issues. i would suggest you test the water (or have it tested at a lfs) to see where you are at in the cycling process. In the meantime, do a large pwc. <Goldfish in a cycling small tank typically need 50% change daily. Some go as far as 100% change daily in a hospital tank - ie the tub-to-tub method.>
Without clean water, you are not going to be able to manage the fish well. Once you ge the water parameters under control, you might consider treating with an antibiotic food if the spots are not getting better. And you will need to seriously consider a bigger tank if you want to keep goldfish long term. <Do you know that your fantail can live up to 7-10 years & get to 8+" under good condition?>
