The initials FGA mean that person is a Fellow of the Gemmological Association of Great Britain and the initials DGA mean that they have passed the Association's Diamond Diploma. They also mean that the person is a member of the Association.
If you would like to check someone's right to use these initials you may check on the Association's FGA and DGA Register here
I have passed both the Association's Diamond and Gemmology Diplomas which involved 3 years of study, course assessments, written exams and practical exams. This was a fantastic starting point, but the world of gemmology is both a vast subject and ever changing, so I continue to study in order to improve my knowledge of the subject I love and teach. Any day that involves working with or studying gemstones is a good day for me!