What is the easiest way to do a nitrogen cycle?

The best and fastest method of cycling is to get some filter media from a well established system. You still need some time, but more like2-3 weeks instead of 6 weeks. Also, common goldfish are inappropriate species for a 6 gallon tank, but that's another answer to another question.

6 gallons is too small for a common goldfish. commons grow 12 inches and need 55 gallon tanks or ponds eventually. in a small tank they become stunted and die. I think the easiest way to cycle a tank is to put a piece of a tilapia filet in the foot of a nylon stocking, throw it in the tank in front of the filter and test the water a couple times a week. very carefree, you don't have to worry about the well being of any live fish and you don't have to measure out pure ammonia all the time.

The easiest way is to get a filter from an established tank and some gravel and you just about have an Instant cycle . In your case you would have to start from scratch with something like a raw dead shrimp or Ammonia . But a Common goldfish cant be kept in 6 gallons . A Common goldfish will grow 1ft and needs a 30 gallon tank then a 55 gallon tank then a 75 gallon Tank. Get a heater and you can keep a Betta fish .