Everyone wants to know what sex their goldfish are. Some people are just curious ("did I give them the right names?") while others think they might like to breed their fish. I'll try to offer a collection of methods I have read about here. However, I'm never totally sure about my own fish until I see them in breeding condition. Until then, they all have names like Pat, Sandy, Terry, Dana ....
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. See some more discussion in the FAQ section.
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.
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.