Start with method as well as content: What Is History? is useful for the problem of historical facts, and The Historian's Craft gives a concise route into evidence, method and historical purpose. For the ancient side, SPQR: A History of Ancient Rome, The Histories, The Peloponnesian War and The World of Late Antiquity: AD 150-750 give different models of narrative, argument, transition and source handling.
For lecture-style breadth, Introduction to Ancient Greek History covers Greek history from the Bronze Age to the classical period, while Roman Architecture and Roman Art and Archaeology connect Roman history to urbanism, material culture and archaeology. To widen beyond Greece and Rome, Superpowers of the Ancient World: The Near East is a useful route into ancient Near Eastern cultures and interaction.
For listening and video, use In Our Time: History, The Ancients, HistoryExtra Podcast and The Rest Is History as starting points for follow-up reading rather than as substitutes for books. The University of Oxford, Oxford Academic, The British Museum and Gresham College channels help connect admissions expectations, academic authors, artefacts and public lectures.
Keeping a short historiography notebook is useful. After each book, lecture or podcast, write down the historian’s question, the evidence used, the argument made and one objection you would raise.