First of all, I think everyone ought to read Xenophon's Anabasis (“The March of the Ten Thousand”) in translation; it is an eye-witness account providing valuable information on Greek, Persian, and Thracian warfare c. 400 BC and reads like a well-written adventure novel; Caesar's De Bello Gallico (“On the Gallic War”) is stylistically indebted to it.
If one wants to read more classics, one could consider starting with Herodotus Histories, Thucydides Histories, Xenophon Hellenica (Xenophon's other works are worth reading too), and Polybius Histories, in that order.
Mary Beard has written a number of books on Roman topics, combining up-to-date scholarship with accessible language; I see you already included two titles, but the others are worth a read as well, and they're affordable ($/£/€ 10-20 range); there is a list at https://en.wikipedia.org/wiki/Mary_Beard_(classicist)#Books
Civilizations do not exist in isolation, they're all part of a continuum, therefore it's worth listing textbooks covering specific subjects extending far beyond our timeframe; (they typically provide lots of references as well). If you have access to a university library or can find a free download on the internet, I'd recommend:
Lionel Casson Ships and Seamanship in the Ancient World (Princeton 1971)
Sarah Iles Johnston (ed.) Religions of the Ancient World (Cambridge, MA 2004)
Philip Sabin, Hans van Wees, Michael Whitby (eds.) The Cambridge History of Greek and Roman Warfare (Cambridge 2008)
Furthermore, if one is interested in Bronze Age diplomacy in the Near East, read the Amarna letters (14th C BC).