Knowing the signs
To understand how goldfish mate is to understand their maturity. Usually goldfish reach sexual maturity nine or so months after birth. You'll notice the males chasing and bumping the females during this time. Males usually show signs that they're ready to mate through white bumps that appear on their fins, known as breeding tubercles. Females usually get a little larger, a sign that there are eggs present.
The mating process
Water temperature is also important during this time. It is recommended the water be at least 18.3 degrees C. During the mating process, the female will spawn her eggs with the male close behind to fertilise them. Next, remove the parents to another tank so that the eggs can hatch without being eaten. Baby brine shrimp are a great addition during this process.
The new additions
The new goldfish will begin to take form in one to two weeks. You'll know the fertilised ones because you'll see little black dots in the eggs. Be advised not to change the water. Algae is fine because it can be considered as a natural food for the fish. Give them about three months and you'll notice colour change. From there, feed them and change the water regularly.