I think your issues are simply that your tanks havent cycled yet and you arent doing enough water changes to keep your ammonia and nitrite at safe parameters. Your nitrate is 0 so you have no cycle happening.
Your water change schedule should be enough to keep your ammonia + nitrite combined below 0.5ppm until you are cycled and they consistently stay at 0ppm. If your tap water is 0.5ppm ammonia that might be difficult, but water changes should be able to bring it down lower than they currently are. Once your system is cycled the ammonia in the tap water shouldn't be an issue. Cycling a tank typically takes 6 weeks but can take much longer. Introducing some filter media from an established filter is the best way to speed this up. Perhaps you have a friend who keeps fish who could let you have some?
If you wanted to use an alternative water source, i would say your options are bottled water, distilled water, RODI
water, or rainwater. I think its Caliban who posts regularly on here who recommends rainwater.