In English, capitalized words are proper nouns- which includes a title, a name, or a specific place. "The" is capitalized because it is the beginning of the sentence and follows standard grammar. "Battle of Bunker Hill" is capitalized because it is the title of an event. "American Revolution" is also capitalized because it is the title of an event. Therefore the sentence should be, "The Battle of Bunker Hill was the first major battle of the American Revolution."
