Gardiner and Gurney's books are hopelessly outdated, but they were superb scholars, and both books are still well worth a read (though not as a first introduction).

The Kingdom of the Hittites and Life and Society in the Hittite World by Trevor Bryce are the best introductory overviews of Hittite history and society in English, but I recommend starting with Warriors of Anatolia, which is essentially a more readable and speculative combination of his earlier books. The Hittites and Their World by Billie Jean Collins is also an excellent introduction to the Hittites.

A History of Ancient Egypt by Marc Van de Mieroop is the standard resource for ancient Egyptian history. Temples, Tombs, and Hieroglyphs: A Popular History of Ancient Egypt by Barbara Mertz is pretty decent and a lively read; the companion volume Red Land, Black Land: Daily Life in Ancient Egypt is excellent as well.

Saggs' Babylonians (not to be confused with his earlier book The Greatness That Was Babylon) is a decent albeit fast-paced overview of Mesopotamian history. I also recommend A Short History of Babylon by Karen Radner.

Fitton's book on the Minoans is still fairly up to date and reliable, but I recommend also taking a look at the recently published Minoan Crete: An Introduction by Vance Watrous. Taylor's book on the Mycenaeans is quite outdated; The Mycenaeans by Louise Schofield is a better introduction.

Markoe's book on the Phoenicians is still pretty decent. A few additional suggestions:

• A Short History of the Phoenicians by Mark Woolmer

• The History and Archaeology of Phoenicia by Hélène Sader

• Phoenicians and the Making of the Mediterranean by Carolina López-Ruiz

u/Bentresh commented on most of the books I'm familiar with already, and as he said they're mixed.

I'll just comment on J.M. Cook's The Persians, mostly because I think its story is kind of interesting. Published in 1983, it was academically dead through no fault of its own. As Cook was writing his book a series of conferences called the Achaemenid History Workshops (1981-1989) were taking and revolutionizing the subject matter. His book is a very interesting study of the Persian Empire with mostly Greek and Biblical sources rendered completely obsolete by the Workshops encouraging the whole discipline to make better use of sources from Egypt, Babylon, and Persia.

It's a bit disappointing because nobody has really succeeded at trying to fill that niche of an easily accessible detailed history of the Persian Empire in decades. The closest equivalent would be a combination of Ancient Persia: A Concise History of the Achaemenid Empire, 550–330 BCE by Matt Waters and A History of Ancient Persia: The Achaemenid Empire by Maria Brosius. Waters' book is very short, but provides a good grounding in the basics of Persian history. Brosius is more detailed, but makes some unconventional choices regarding terminology and emphasis.

A good rule of thumb is to try to read something written within the last 10, or 20, or maybe 30 years when possible, in dealing with archaeological subjects. Anything older can't have incorporated more recent discoveries, and won't represent the most up-to-date knowledge on the field. If you can't find something recent, then look for when the author of said book was most active - for example, Lord William Taylour has excellent credentials and experience but his latest experience was in the 1960s, and thus he won't have known of some of the more important discoveries that came later (he died in 1989). Glen Markoe, on the other hand, was born in 1951 and was still active until his early death in 2012 - so his work is more useful, though of course will sadly lose relevance in coming decades.

There are, of course, exceptions, and I'm thrilled to see that Barbara Mertz's books are still recommended (by u/Bentresh, above) though she sadly passed away in 2013!

Another thing to keep in mind is that the questions scholars ask of the past are reflections of the lives we live today. Just as one example, in my field (classical archaeology), we started to ask questions about ancient social networking only within the last 15-20 years, partly owing to the extreme connectivity of the modern world - questions about who was in whose 1st, 2nd, 3rd, etc degree of connection wasn't often a part of discussion in the mid-20th century. So, if I' pick up a book published in the 1960s, in addition to not taking into account discoveries made in the latter half of the 20th c/21st c, the viewpoint is simply different than today's, making for less satisfying reading. But, again- there are always exceptions, so asking in a forum like this or seeking out scholarly reviews are good strategies.