To keep a fish tank, you need to have a good grasp on the nitrogen cycle. This video is the best I've found for explaining how it works to new hobbyists, he really breaks it down in a clear, easy to understand way
You're quite early on the cycling process, so as you've discovered since one passed away, it's crucial to keep on top of testing and water changes, because any reading of ammonia or nitrite, even a low reading, is harmful for the fish. You also have to bear in mind that even if a low reading is survivable - which it isn't always - that the reading will continue to climb without a water change.
I'm not sure what you should do regarding the 20ppm nitrates straight from the tap, essay will have better ideas. But if it were my tanks, I would get a lot of fast growing plants, ammonia consuming plants like swords, hornwort, elodea, water wisteria and sessiliflora, both to help cycle the tank, and to use as much of the ammonia as possible before it's turned to nitrites and then nitrates. 20ppm is about the max I'd want in my tanks, and that's the level you're at immediately after a water change. I think it's still considered 'safe' for fish up to 40ppm, so if you're stuck using the tapwater, I'd aim to prevent it going over 30ppm, and live plants are one of the best ways to do that.