Picture and procedure taken from cornell.edu

It is a very good idea to castrate male goat kids that are going to be used as pack or driving goats or pets. Castration is best done with two people. One of you should get behind the goat and gently set it on its tail. Grasp the goat's front legs in each of your hands and wrap the front legs around the back legs at the knees. Hold on to the legs at this joining point.

The other person feels the scrotum (loose sac of skin holding the testicles) and locates the two cords. Push one cord over to one side of the scrotum. This is the first cord you will crush with the Burdizzo.

The burdizzo is used on only one side of the scrotum at a time. This is because if you clamp the burdizzo over the entire scrotum you will cause the scrotum sac to gangrene and complications could arise if the scrotum is large.

Clamp the burdizzo over the cord on the side of the scrotum you are doing first. Be careful to leave the goat's teats above the crushing point.

Leave the burdizzo shut ~25 seconds. You can tug on the cord to see if it feels ruptured. You can move the burdizzo down slightly closer to the testicles and open and close it again to be doubly sure that the cord ruptures.

Locate the cord on the other side of the scrotum and position the burdizzo over it.

Close the burdizzo and repeat your procedure.

When you are done, you may see a mark on each side of the scrotum. Your goat may be sore and move slowly for about a day. Be sure to encourage him to move around. At first his scrotum will swell up a little, but the testicles will gradually shrink over the next few weeks leaving a small scrotum sac.
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If you don't have the burdizzo you could use scalpel and forceps. The procedure is almost the same with castration with piglets.