How do you become a gem cutter?

There are some courses available, but by and large teaching yourself is the way to go. There are many books and websites on the topic, and there are gem and mineral clubs around the world which may have members willing to help you get started. Cutting cabochons is very simple, requiring practice and patience. Faceting stones is a bigger investment in machinery, but anybody with decent hand-eye co-ordination and lots of patience can master it.