Keeping the water clean and clear of ammonia and nitrIte is extremely important to the health of any fish, but I just wanted to add that from my personal experience Neons are more sensitive to nitrAte levels in the water than many other fish. I've found that it's better to do more frequent, smaller water changes with them so that the nitrate level stays low enough not to be harmful, but doesn't fluctuate very dramatically, either. For example, going from 30ppm of nitrate down to 10ppm after a water change over and over again will be much harder for their systems to tolerate than staying around 20ppm fairly consistently. Once the bacteria catches up to the bioload and the cycle is complete, the nitrates will usually remain steady, so you won't have to worry about it as much, but a cycling tank is a dangerous thing for a fish, and the littler ones seem to have a harder time getting through it. It sounds like you're getting close to the end of it, if your nitrates are spiking - so hooray for that! I just wanted to share my experience, hope it helps, and good luck on the new tank!