Don't feel bad, vast numbers of us started that way. I started a very long time ago with the same kind of carnival goldfish, aka feeder goldfish, that were a dollar a dozen. Throw them into a 10g tank, and after a week, almost all were dead. But they were only a buck a dozen, so back I'd go to get more. Every three months, put whatever fish were still alive in a bucket, completely empty the tank and sterilize it, put the fish back, repeat for about a year before I gave up. Now I know better. Live and learn.
No, you should still cycle the tank no matter what kind of livestock it has in it. If you can, look for a product called Biospira. Must be kept in the store's fridge. Don't let them talk you into Stress Zyme, Cycle, or any other crap in a bottle. ONLY Biospira has the correct LIVE bacteria, the rest contain dead bacteria that do nothing but add to the already heavy bioload. If you can find Biospira, you can add the fish right away. Otherwise, you'll need to cycle by other means.
Oh, and not sure if that was just a mis-statement but coldwater aquariums are freshwater aquariums.
There's just a differentiation between coldwater aquariums and tropical aquariums, where tropical employ warmer temperatures.
BTW, welcome!
Oh, and a final general rule, very, very few store clerks have any idea how to properly take care of fish, so they rarely know what they are talking about. Relying on any advice they give is usually not a good idea. Researching and asking those with experience is your best bet.