Technically, you don't really have to have any specific education or experience in order to become a diplomat; you just need to pass an extremely rigorous series of tests, written, oral, and physical.
Prior to even attempting that test, you should probably have, at a minimum, a bachelor's degree. Most diplomats working today have degrees—bachelor's, master's, even doctorates—in economics, law, political science, foreign relations, geography, commerce, medicine, and even international history.
Because diplomats can have a variety of jobs—translators, managers, economists, to name a few—if you find you want to specialize in a certain area of diplomacy, it's a good idea to get an education in that area. For example, a degree in science or even medicine for someone interested in healthcare administration is preferred and displays their commitment to their career trajectory as a diplomat.
Many diplomats have already traveled abroad—by studying there or working there—furthering their interest in their career of choice.
After you've obtained your degree and experience, it's time to take that Foreign Service Exam, which consists of a written and oral test plus a negotiating drill and a series of physical tests as well as a deep look into your background. (Apparently these are quite grueling and tend to weed out those really interested in becoming a diplomat and those really, really interested and committed.)