So, you're thinking to yourself, "How do I remedy my lamentable ignorance about all things Byzantine?"
Hopefully you aren't as badly off as the poor BBCer who referred to "Manuel Paleologus II, Emperor of Byzantine," which was enough error in six words to give me the bends.
The last is easy, obviously. But the name order is another major screw up. It's Manuel II Paleologus because the first Manuel was from a different dynasty, the Comneni. That Manuel is properly referred to as Manuel I Comnenus. A little thing, but it betrays a rather serious problem: too few people in the West have even the faintest knowledge of the first Christian state, which both shielded Western Christendom and was responsible for the glorious civilization of the Christian East. It is an essential part of our patrimony, but it is a universal that Byzantium gets glossed over quickly. Which, considering it lasted approximately twelve centuries, is inexcusable.
Fortunately, dear reader, you've come to the right place, so long as you are willing to take the first step and admit you have a problem.
With that in mind, here are some book recommendations:
A History of the Byzantine State and Society by Warren Treadgold. At the risk of punnery, it is the gold standard of one volume histories. Yes, it appears to be doorstop thickness, but the 800 pages of actual history (the remainder is comprised of a valuable annotated bibliography, along with endnotes and an index) seems short by the time you finish. It covers religious issues with a deft touch (you'll have the Christological controversies nailed by the end of it), devotes essential space to economic issues and offers valuable windows on Byzantine literature. Well-written and willing to offer new interpretations, it is essential.
History of the Byzantine State by George Ostrogorsky. The standard before Treadgold, it's still valuable. Focuses more on political developments, but does so capably. Has the added bonus of still being in print.
Byzantine Art by Robin Cormack. From the Oxford History of Art series, this is an excellent survey of the various forms of Byzantine art and architecture.
The Glory of Byzantium. Another artistic survey, this one covers the period of 843-1261 (from the end of iconoclasm to the liberation of Constantinople from the rotten Latin Empire). Beautiful plates, and excellent introductory articles make it invaluable. If nothing else, it's one hell of a coffee table book.
Rulers of the Byzantine Empire. By Kibea, a Bulgarian publisher, it's a beautifully illustrated encyclopedia of the most significant figures to become Emperor. The translation is a little creaky in spots (e.g., uses "scientist"/"scientific" instead of "scholar/scholarly"), but is otherwise good.