This is the opposite end of things--a recent, updated/revisionist (not a bad word) perspective, but:

/u/shlin28 and I are both, in our spare time, poking at Anthony Kaldellis' The Byzantine Republic: People and Power in New Rome. Kaldellis, building on a long history of earlier publications, is pushing hard but smartly against the dominant historiography of a quasi-absolutist/theocratic monarchy in the East. "Republic" and "people" are the key words in his title; he reconstructs a much more complex Byzantine--Roman!--government and citizenry.

It's definitely good to have a grounding in the major older works in the field, but don't skip this one.

I have to begin with a word of caution regarding reading early histories of Byzantium, as their interpretations are often problematic and may give you the wrong impression of what the Byzantines were like. As noted by /u/sunagainstgold, major revisions occur all the time in this field. Kaldellis' Byzantine Republic (2015) is an excellent example. It's a great read and reviews generally praise his refreshing take on the 'Byzantine' state, but it did not cover all the factors involved (the socio-economic aspect most prominently) and it will take a long time before a new 'consensus' can emerge. Nonetheless, I suspect that his description of how 'Byzantine' succession and political intrigue worked is more accurate than what historians like Bury suggested a century ago, which is why I recommend starting with more recent material first. Historians have made huge strides in recent decades, with the most obvious being the use of the archaeological evidence, which sometimes provides interesting confirmation, or indeed refutation, of what we can gleam from the literary sources. A fine discussion of recent developments can be found in Byzantine Matters (2014), a short book by the eminent Byzantinist Averil Cameron, who rightfully points out how the things we associate so fundamentally with Byzantium, whether its firm adherence to orthodoxy or its attitude towards art and icons, should nevertheless be revisited.

Classics are of course still useful, but I generally don't see Bury being cited all that much in present scholarship. George Ostrogorsky's History of the Byzantine State (1958/1968) is more common, but it's narrative is for me problematic, particularly for the seventh century, which is what I work on. Ernst Stein's Histoire du Bas-Empire (1949) is more frequently cited and has more interesting insights, but it's probably less accessible for most people. A book that does still stand the test of time from this period is A. H. M. Jones' The Later Roman Empire, 284-602: A Social, Economic, and Administrative Survey (1964), which is still the go-to book on the late Roman bureaucracy. A few years after Jones' book, Peter Brown published his magisterial The World of Late Antiquity: From Marcus Aurelius to Muhammad (1971), which quite literally reshaped the field. By taking a cultural perspective on a field that was still haunted by the ghost of Gibbon, we instead learn from Brown about the cultural vitality of the late antique world, of how despite the crises of this era, we can still view this period in its own terms, to see forces other than 'decline' being at work during the 'collapse' of the Roman Empire. This tendency continues to shape the field, which for me is a reason to look at Brown first rather than the older work of Bury.

Brown's work would, for me, be the ideal starting point for anyone interested in Byzantium, but it only covers the first few centuries. For a more long-term view, you can't go wrong with one of Judith Herrin's Byzantium: The Surprising Life of a Medieval Empire (2008), Averil Cameron's The Byzantines (2009), or Jonathan Harris' The Lost World of Byzantium (2015) as an introduction. A more bulky choice would be Warren Treadgold's A History of the Byzantine State and Society (1997), which presents a decent narrative despite its length, whilst a more in-depth look can be found in The Cambridge History of the Byzantine Empire (2008), as each chapter is written by an expert on that specific topic. Shorter chunks of time are examined by Peter Sarris' Empires of Faith: The Fall of Rome to the Rise of Islam, 500-700 (2011), Mark Whittow's The Making of Byzantium, 600-1025 (1996), and Michael Angold's The Byzantine Empire, 1025-1204: A Political History (1997), all three of which are the 'standard' introductory texts for students studying their respective periods.

I work on the earlier period and so I can recommend a few other books as well, if only to give you a flavour of how much has changed since the early twentieth century. The famously 'Byzantine' controversy, Iconoclasm, has for instance been questioned in recent years, particularly by Leslie Brubaker and John Haldon's lengthy Byzantium in the Iconoclast Era, c. 680-850: A History (2011), in which they argued that 'Iconoclasm' was a rather more political affair and that some 'Iconoclasts', for example Emperor Leo III, were not against icons at all. A briefer and more accessible sample of their argument can be found in Brubaker's Inventing Byzantine Iconoclasm (2012). Their views are not necessarily agreed upon by everyone, but their arguments have certainly shaken up the field and makes it clear how much has changed, even in things as basic as the narrative.

Haldon has also written about the crises affecting the seventh century in his The Empire That Would Not Die: The Paradox of Eastern Roman Survival, 640–740 (2016) and it is remarkable how much has changed just since the 1990s, when Haldon wrote another book dealing with the same topic. His description of the religious controversies are now made more nuanced and the narrative informed by recent studies, most prominently (for me) Phil Booth's Crisis of Empire: Doctrine and Dissent at the End of Late Antiquity (2013). New retellings of decisive events, such as Marek Jankowiak's redating of the siege of Constantinople in the 670s to 668 instead , are also taken into account - something that you simply won't read about in anything published before 2013. Most dramatically though, Haldon incorporated a huge amount of scientific data in his new study. This even extended to the pollen-count in seventh-century Asia Minor and data taken from ice-cores in the Arctic, which tells us, for example, that just as the empire was facing its greatest crisis, Asia Minor became more productive agriculturally, giving Constantinople a slight edge after the loss of its most wealthy provinces. Understandably, these insights could only be gained in the twenty-first century and would not have been included in any study published before then.

This is only a very brief overview of new developments in Byzantine studies, but I hope I have made the case that it is worth exploring some more recent publications first, as opposed to starting with Bury and his colleagues. If you have any questions, please let me know and I'll be happy to help.

In addition to what others have mentioned, I would also recommend "Hellenism in Byzantium" by Anthony Kaldellis. For primary sources, I would recommend, for the middle period, the History by Leo the Deacon (which can be found [here] (http://terpconnect.umd.edu/~sullivan/LeoTheDeacon.pdf)), the History by Micheal Attalieates, "Chronographia" by Micheal Psellos (which can be found [here] (http://sourcebooks.fordham.edu/basis/psellus-chronographia.asp)), the "oration of admonition to an emperor" by John Kekaumenos (which can be found here), and the Alexiad by Anna Komnenos (which can be found here).

Edit: since you're also interested in military history, I would further recommend "Sowing the Dragons teeth" by Eric McGeer, which is a translation of two Byzantine military manuals and a commentary on both the manuals and byzantine warfare in general during this period (note that he does make the odd decision of translating "Roman" as "Byzantine" in the manuals).

I am not a specialist in that area, but I have had coursework on the Byzantines. Deno John Geanakoplos created a collection of documents called Byzantium: Church, Society, and Civilization Seen through Contemporary Eyes. The contents are organized by subject and chronological order and had some context and explanation given for the included documents. You might also be interested in Michael Psellos' Chronographia, a collection of biographies about the emperors and empresses from 976 to the 1070s written by a scholar that lived under a number of their reigns. I loved the books, but I'm not sure if collections like these are what you are looking for, and to reiterate, I am not an expert in the field, so if more qualified people would dispute this recommendation, follow their guidance and not mine.

Given the importance of the military in preserving the empire, I think this would be good to read, Warren Treadgold's Byzantium and It's Army, 284-1081.

Specifically regarding the final fall of Constantinople, you could not look for a better book than Steven Runciman's The Fall of Constantinople 1453. Runciman is one of the foremost scholars of Byzantium and is generally one of the people credited with bringing Byzantine studies back to a western audience.

His History of the Crusades is excellent too, though it only deals with Byzantium in relation to its role of involvement in the Crusades and with the Crusading state, and of course the disastrous 4th crusade.