General introductions which are made for a non-specialist audience:
Soldiers and Ghosts by J.E. Lindon, an excellent summary of the mentality and social aspect of war in ancient Greece and Rome.
Mediterranean Anarchy, Interstate War, and the Rise of Rome by Arthur Eckstein, an examination of the rise of Rome from the point of view of the field of International relations.
The Greeks Overseas by John Boardman, an archaeological and cultural discussion of Greek colonies across the Mediterranean in the period before Alexander.
The Rise of the Greeks by Michael Grant, covering Greek history in the archaic history. One of the most in-depth examinations of the Greek world as a whole that's available to non-academics, rather than just the major players.
SPQR, a general history of Rome from its beginnings to the end of the Empire by Mary Beard.
Rise of Rome by Kathryn Lomas, which focuses on Italy from the bronze age to the Pyrrhic Wars and tries to place Rome in a wider Italian context.
For those with cash or prior background in the period,
Alexander to Actium by Peter Green is a frighteningly long and comprehensive tome covering the entire Hellenistic period from the death of Alexander to the Battle of Actium. It is an extraordinary and magisterial work and there's really nothing like it in the classics.
I don't actually have any one book to recommend for classical Greece specifically. Unfortunately there aren't any big surveys of the period and I haven't had the money to buy more classical history books.
Finally, for a purely visual tour of Classical Greek cities, I am going to no-joke recommend Assassin's Creed Odyssey. It is lacking in historical accuracy in a lot of ways, given it's a video game, but the way that it brings the world of the period, and especially the
city of Athenai to life is nothing short of breathtaking. There is simply no better visual representation of Classical Greece as it was than Odyssey. I haven't even bought the game yet since I can't justify spending 80 bucks on it, but I will watch people explore its cities on youtube if I ever need inspiration.