The best introductory work to the Middle Ages that I've found so far would have to be Introduction to Medieval Europe 300–1500, by Wim Blockmans and ‎Peter Hoppenbrouwers. It's well illustrated with maps, diagrams and relevant photos, up to date with the latest research, comes with a useful companion website and covers the whole period from Late Antiquity (300 AD) to the beginning of the Early Modern period (1500 AD). To some extent it fails your criteria, as the military sphere is explored very little and social and religious spheres are explored in considerable detail, but this is an artifact of modern scholarship. John M. Riddle's A History of the Middle Ages, 300–1500 probably has a better overview of politics than Blockmans and Hoppenbrouwers, though otherwise they're more up to date, but social and religious elements of society are still given quite heavy weight and military matters are relegated to the background.

Even going back a quarter of a century to David Nicholas' The Evolution of the Medieval World: Society, Government and Thought in Europe, 312-1500, which has a very strong focus on political history, religion and society are given overwhelm preference to military history. The truth is, most medieval scholars aren't particularly keen on medieval military history, except for where it briefly intersects with society. There's quite a long history here with reactions against scholarship of the 19th and early 20th centuries and the often amateur, battle focused nature of early works on the subject, and even today I think many scholars still don't quite take military history as seriously as it should be - although this is changing as trained academics have largely taken over the subject and expanded beyond the narrow confines of battles and technicalities. In any case, it's an unfortunate truth that no modern textbook is going to provide much of use with regards to military history.

Regarding military history, there's really not a single volume I can recommend. The broadest overview would be David Nicolle's Medieval Warfare Source Book, which attempts a global overview of warfare between 400 AD and 1400 AD in two volumes. It has the best coverage of any book on medieval warfare, though Asia deserved to be its own volume, but the limits of the format mean that you only get a bare bones view. Nicolle does a splendid job of providing the crucial details, but the short length and style of book don't allow analysis and expansion on his points.

The most recent book on medieval warfare as a whole is Bernard and David Bachrach's Warfare in Medieval Europe c. 400-c.1453, although unfortunately the authors overwhelmingly stayed within their comfort zone so that most of the book is focused on the period of 700-1200 AD and England, France and Germany are the primary focus of their attention. The Bachrachs also hold some, shall we say, controversial views on medieval warfare, which come through strongly in their work. It is useful for their discussion on the logistics of war and methods of studying medieval military history - two areas in which both authors have contributed immensely to the field - and their perspective on the nature of early medieval warfare shouldn't be dismissed out of hand, either. That said, if your interest lies in the later middle ages or outside of England, France and Germany, it won't be the biggest help.

Outside of these two works, the best works on medieval warfare are volumes on specific periods. Guy Halsall's Warfare and society in the barbarian West, 450–900 is the best volume on early medieval warfare, while John France's Western Warfare in the Age of the Crusades 1000-1300 covers the High Medieval Period extremely well. Both do tend to exclude Northern and Eastern Europe, but this gap can be partially filled by Philip Line's The Vikings and Their Enemies, which provides some information on Celtic, Scandinavian and Slavic warfare in the period between 750 AD and 1100 AD.

The later middle ages are, unfortunately, sadly neglected at the moment with respect to syntheses at the moment, both of warfare and of society as a whole. If you're interested in the period 1300-1500 AD, then the relevant chapters of Philippe Contamine's War in the Middle Ages, old as they are, are still the best option available at the moment.