Reddit mentions of The Oxford History of Byzantium

This is a fun little comic, but it creates a few misconceptions about Eastern Roman history:

• Western Germanic kingdoms were relatively Romanised and before Justinian's reconquest, many enjoyed friendly relations with the East, particularly the Vandals. In fact, arguably Justinian only invaded Africa because some Vandal nobles overthrew the pro-Roman king Hilderic.
  
  
• It implies that Justinian was an able ruler till his death - unfortunately, whilst he was definitely an ambitious ruler, by the end of his reign he was exhausted and ruled more as a tyrant than the vigorous young emperor who codified laws and launched reconquests. A contemporary poet noted that 'the old man no longer cared' when summarising his later reign. In the 550s, the Persian war had dragged on for decades, the Balkan frontier suffered yearly raids and in Italy, the Lombards would soon invade and destroy whatever stability Justinian restored. There were many conspiracies against the emperor, a few even implicating Belisarius, the hero of the Vandalic/Gothic Wars. By the end, Justinian may have even embraced a form of the miaphysite heresy...
  
  
• Justinian's successors were not incompetent, well, Justin II possibly (he allegedly went mad after an important fort was lost to the Persians...), but Tiberius II was definitely an able general who persecuted the Persian War quite well. The comic also missed out Maurice, who again was a great military leader who made massive territorial gains against the Persians and defeated the Avars on the Danube frontier. The usurper Phocas also stablised the Balkans and did not lose that much ground against the Persians, in fact Heraclius' revolt probably led to the collapse of the eastern armies, rather than vice versa.
  
  
• I also don't think Phocas was unpopular because he persecuted Christian sects, since all the emperors did that, with Justinian and Heraclius being two great examples. Phocas was even quite popular with SOME people, such as Pope Gregory the Great and the Blue circus faction.
  
  Still, the comic is a great way to get people interested in history, and if anyone would like to read a bit more, here are a few good general introductions :)

Unfortunately, there are some who are better suited to recommend some of the ancient works that you are looking for, but here are some for the classical/late antique period:

On Persia:
Parveneh Pourshariati's Decline and Fall of the Sasanian Empire was extremely important and full of new approaches to Persian history in the centuries immediately prior to Islam. It's not the easiest of reads from cover-cover, but if you are seriously interested, go here. It is especially useful for its discussions of some of the "minor" pre-Sasanian and Sasanian families and dynasties that made up Persia and its environs.

For a very similar period of Persian history is Touraj Daryaee's Sasanian Persia: The Rise and Fall of an Empire

On Syria:

Ross Burns' Damascus: A History is an extremely useful and extremely readable history of the capital of Syria as a case study of settlement in Syria over the millenia. The early sections of it I would highly recommend.

More generally:
Cyril Mango's The Oxford History of Byzantium is a useful, concise work on Byzantium.

It hasn't been mentioned here, but Peter Brown's World of Late Antiquity is also considered by a great many to be essential reading on the period, if you've never read it before. Plenty of people strongly disagree with it, but everyone who works on the ancient/late-antique world is forced to engage with it at some point, whether they like it or not. It is also extremely readable.