I minored in Ancient history in undergrad, and, in my more honest moments, I admit I probably should have majored in it. Not that I'm dissatisfied with political science, it's just that in undergrad I spent far more time discussing/reading/sitting in class and learning about the Ancient world than I did any single political topic (and possibly more than all political topics put together, though I haven't gone back and looked at my schedule to try to figure it out). Even now I haven't given up my delight in the Classical world, and as soon as I'm done flogging the politics out of the good Reverend Edwards, I fully intend to get back to my first love.
So, with all that having been said, this is a list of books from or about the ancient world that 1) I've read myself; 2) I wish more people would read; 3) aren't so difficult that they couldn't just be picked up and read by anyone, whether they've got a history background or not. To simplify things, I will not be including philosophy, religion, or fiction on this list. History only!
1) Twelve Caesars by Suetonius. This book is simply a must-read. Suetonius was some form of minor functionary in the Roman government, but the kind of bureaucrat that got access to the Imperial archives. Which meant that he could read all about all the terrible things that Roman Emperors had done, and write about them (after he got caught and got fired, and had some leisure time on his hands). Suetonius tries his hardest to give each emperor a fair shake, listing first the good things they did, and then chronicling their much more numerous vices.
2) Campaigns of Alexander by Arrian. Arrian was one of the first military historians, and is the primary source (along with Plutarch, see below) for our knowledge of Alexander the Great. While Arrian isn't necessarily a spectacular writer himself, he's interesting enough to keep the narrative flowing. Besides, Alexander the Great is the single most interesting figure in the ancient world, and it's almost a given that a book about him is going to be fantastic.
3) Parallel Lives of Noble Greeks and Romans by Plutarch. Plutarch was a philosopher and pagan priest who wrote a great many books, all of which are worthy of attention (his essay on "Talkativeness" is fantastic, he argues that talkativeness is a sin because it can never be satisfied), but his biographies are really the place to start. In the Parallel Lives, Plutarch pairs up Greeks and Romans, comparing their accoplishments and commenting on their similarities and differences. Cicero, Demosthenes, Caesar, and Alexander the Great all make appearances. Even better, these biographies aren't written as a narrative whole. so you can go through and pick out the ones you want to read and pass over the rest, without missing anything.
4) Rise of the Roman Empire by Polybius. Polybius was a Greek prisoner of war being held by the Romans, and had a front row seat to the destruction of Carthage. He wrote this history to explain to the Greeks why it was that Rome was the up-and-comer in the world, while their own cities languished or even collapsed. Mostly dealing with Hannibal and the Punic War, this is one of the most readable books from the ancient world and is well worth a look, especially book 5, which covers the battle of Cannae, and book 6, which describes the Roman constitution.
5) The Catilinian War by Sallust. A governor of North Africa, Sallust was a friend of Caesar and both spectator and participant in the Roman revolution. In The Catilinian War, he writes about the conspiracy of Catiline to overthrow the Roman government (just a few years before Caesar managed to throw it off). This short book is useful both as an interesting narrative, and as a great way to meet such famous characters as Cato, Cicero, and Caesar.
6) The Spartans by Paul Cartledge is the best introduction to the warrior city of ancient Greece. Both readable and informative, Cartledge lays out the structure and culture of Spartan society, as well as discussing its growth, apex, and destruction.
7) Civilization Before Greece and Rome by H.W.F. Saggs is a good catch-all for the "everything else" of the ancient world. Discussing Egypt, Assyria, Babylon, Persia, the Hittites, and so on, Saggs gives an engaging overview of the development of society prior to the rise of Greece. The economics chapters can bog down a bit, but in general this is an excellent read.
8) Gods, Graves, and Scholars by C.W. Ceram is a survey of the greatest hits of archaeology from the beginning up through the 1940s, and, well, I'm glad I didn't read this until after I'd graduated college, because I would have seriously considered converting away from political science and into archaeology. Unfortunately, archaeology these days is much more about digging through the dirt with a toothbrush and much less about fighting off bandits while you dig up gold. Fortunately, this book is about the earlier stage in archaeology...
9) Alexander the Great by Robin Lane Fox. There are many great biographies of Alexander (both ancient and modern), but this is one of the best since it not only tells Alexander's story, it explains the ancient world in a way that is understandable to the modern reader. There are others that could be read, but this is probably the one to start with (my favorite is Wilcken's, which was written in Germany in the 1930s...). Fun fact about Lane Fox: he was the historical advisor on the Oliver Stone Alexander movie (which is one of the reasons it was historically accurate, even if it's a terrible movie). As part of his payment for putting his name on the movie, he demanded that they let him take part in a cavalry charge (apparently that's a common wish amongst Alexander scholars), and at one point, so the rumor goes, he got so into it that they had to stop shooting because they were afraid he was going to spear someone...
10) The Battle That Stopped Rome by Peter Wells. This isn't an earth-shattering book or anything like that, it's just a solid example of how one historical event can be covered in an interesting way with a full-length book. I could have easily picked one of many others, but this one is fairly short and covers an important battle.