The culture of the time was a feudal system, and Beowulf was born into the Lord/Warrior/King line. His training would have been the same as any other peer. The poem itself does not detail Beowulf's younger years except the one episode brought up by Unferth as an attempt to minimize Beowulf's great strength.
What set Beowulf apart was not his training, but rather his character, his integrity, his loyalty, and his strength. He has all the characteristics of a true epic hero-and maintains them from his youth (killing the sea monsters) to his old age (engaging the dragon.) Those qualities (integrity, strength, loyalty, selflessness, generosity) cannot really be taught through training, but are "caught" by watching good role models and having an internal sense of morality.