Read.

Read the primary sources themselves, yes. It helps to have a good idea, mentally, of the actual rhythms of their language, the format of the work, the limits of what the text actually says on the surface.

But from a historian's perspective, at the beginning, read those secondary sources. Read the people who "can just drop names and analysis like it's nothing." Read their footnotes. Pay close attention to what sources they're using for what pieces of evidence. Crucially: pay attention to how they're using each source. How does the scholar grapple with the author's goal for the text affecting its content? What topics is a source usually called into play on?

When I'm taking notes on a book, I always start by writing down the book's thesis, the historiographical background (the research it builds on or disagrees with), and its major body of sources. That last one is really helpful to note, especially if you're consciously trying to be familiar with them as well as with the narrative of the past.

For systematic acquisition about a topic independent of classes, my favorite strategy is to follo as closely as possible a PhD comprehensive exam reading list. Ancient/classical history is hard because a lot of Google results are for Greek and Latin primary sources, but here's one from Michigan and one from Arizona that aim at sweeping overviews in secondary scholarship.