The nitrogen cycle is the buildup of natural bacteria that convert fish waste (ammonia) into less-harmful nitrites, and then the buildup of more natural bacteria that convert the nitrites into a much-less-harmful nitrates.
The best way to avoid illness during this period is to not have fish in the tank while it's happening. Since it's too late for that with your tank, I would suggest following all of the details in "Cycling with fish" in the following link.
http://www.fishlore.com/NitrogenCycle.htm
