I found this information with a picture at www.geocities.com/Tokyo/4468/fishsex.html
The classic way to sex goldfish is based on the sexually dimorphic traits they develop when they are in spawning condition. At this time, females become heavier bodied in the abdomen (and purportedly a bit asymmetric). But the big change is in the males; they develop white pimples, called "breeding tubercles", on the gill operculum and along the front rays of their pectoral fins.
I've noticed that males that have been through a few spawning seasons seem to retain roughness on their front pectoral rays. Usually, when someone shows me a proven pair of goldfish it is pretty obvious which is the male and which is the female. But when I try to judge the sex of my fish, I often have little confidence in the decision. I think there is a gestalt to it. I will list some of the indicators I have heard of or read about for goldfish sexing outside of the breeding season. Incidently, I would not expect any of the methods discussed here to work reliably for fish prior to sexual maturity ... say a year.
Midline ridge: Males develop a ridge on their undersides that starts in back of their pelvic fins and ends at their vent opening. The ridge is absent or much smaller on females. (N.B. I have observed females with impressive ridges. Hmmm)
Firmness of Abdomen: For my heavy-bodied goldfish, the area between pelvic fins and the anal fins is fairly pliable on the females. For the males, this area is usually very firm.
Vent shape: According to several books, you can sex goldfish based on vent shape. I haven't been able to figure this out yet, so I won't confuse with an attempt at explaining what I have read. However, another hobbyist, Red Arthur, has used it and he supplied the following sketch to illustrate the method. The sketch also shows the location of the breeding tubercles on the males. Thank you, Red!
Pectoral fin shapes: For short-finned varieties, males have longer, more pointed pectorals with a stiffer leading ray. Females have more rounded pectorals and shorter, finer front fin rays. For long-finned breeds, the fin shape is hard to judge. But the first ray length and thickness should still be somewhat different. This works pretty well.
Other features: Some other characteristics that I think would be very hard for beginners to make use of: males are alledgedly brighter colored, more active, and have longer fins.
Using a known female: Al Foster suggested introducing a fish known to be a female into a tank of goldfish of unknown sex. He says that the males will all "check out" the new female, while the females will show no interest.
Probably the best way to learn to sex goldfish is to try to get opportunities to look at adult fish of known sex. The next best is probably to try to find some place (petshop?) that has an assortment of goldfish with bodies longer than three inches. And then just try to guess their sexes, based on the indicators above, to get experience looking at goldfish.