Very little, I'm ashamed to say.
Most of what I do know is recent history of the past 100 years or so. Of course, everyone learns about Princip and the other Bosnian Serbs that assassinated Franz Ferdinand and his wife, leading to WW1. The most famous Serb scientists/engineers in the US are Tesla and Pupin; Pupin has a building named after him at Columbia University in New York City, and there is a Tesla museum being worked on (on Long Island), where he had a laboratory.
I heard of the Black Hand, and studied some Yugoslav history, including especially Tito (although he was not a Serb, the capital was in Belgrade), interesting because he often defied the Soviets, and I think did not permit Soviet troops on Yugoslav territory.
And of course recently, Milosevic, the Kosovo War, etc.
Most of the earlier history of the Balkans is very complicated, and I don't know much about it, beyond the long Turkish occupation.