I actually looked into this specifically many years ago, it's a long story as to why I wanted the woman who is now my ex to take a particular course. The best and most reputable source for education leading to a career in the diamond trade is the "Gemological Institute of America".
They will provide an education that will open doors for you in the future. Store owners want knowledgeable people who have taken the GIA course, or its equivalent, to handle their customers. People who buy diamonds and other gemstones expect qualified salepeople.
Wikipedia says this about the GIA:
"...The Gemological Institute of America, or GIA, is a non-profit institute dedicated to research and education in the field of gemology. The GIA is also well known for its gem identification and grading services, and developed the famous "four Cs" (Cut, Clarity, Color and Carat weight) methodology of grading diamonds. Although the GIA is best known for its activities in relation to diamonds, the institute is also involved with colored gemstones and pearls. The educational arm of the GIA confers the Graduate Gemologist diploma upon its graduates; the institute also offers other diploma programs. The GIA was founded in 1931 by Robert Shipley in Los Angeles, California. Today, the institute is headquartered in Carlsbad, California, north of San Diego..."
If you are really serious about pursuing a career in this field, you owe it to yourself to look into the GIA's educational programs.