The best general history of Christianity that I've read is definitely Diarmaid MacCulloch's A History of Christianity: The First Three Thousand Years. It's over 1200 pages, but it's far from bloated. The topic is just so massive. Were there any specific ecclesiastical periods that you wanted more texts on?

I'm afraid I don't know much about the available texts on Byzantine history. The caesaropapist Byzantine Orthodox Church is covered in MacCulloch's overview text, but obviously other texts go into greater detail.

For general histories of the Byzantine Empire, try:

Warren Treadgold, A History of the Byzantine State and Society (Stanford University Press, 1997)

Robert Browning, The Byzantine Empire (Catholic University of America Press, 1980, rev. ed., 1992)

Those are two recent ones that cover the whole empire - Browning's is fairly short, while Treadgold is much longer and packed full of details. Writing a history of the entire empire is a pretty gargantuan task, so it doesn't happen very often, although there are also a lot of good books about specific periods.

There are also these two much older books, classics in the field but probably pretty outdated now:

A.A. Vasiliev, History of the Byzantine Empire (University of Wisconsin Press, 1952)

George Ostrogorsky, History of the Byzantine State, trans. Joan Hussey (Blackwell, 1968)

Though I am not an expert, I would second what /u/WelfOnTheShelf said about Treadgold's A History of the Byzantine State and Society. I used it extensively in a term paper to wrap my head around general Byzantine society for a for a 400 level survey undergrad course, but I would also suggest Adrian Goldsworthy's How Rome Fell for greater context on the origin of the East/West split, which becomes pretty fantastically important once the West "falls".

He sadly goes on a bit at the end of the book comparing the fall of Rome to the present day, but I think he has some interesting takes on the conventional narrative many of us are familiar from popular understanding, and it can help contextualize the state of the Roman world at the time Christianity really started to transform the empire just before Constantine I and after. I would just skip the bit where he gets political.

I think you're barking up the right tree focusing on Christianity. My professor (John Birkenmeier) was fantastic, and his class hammered home the idea that you cannot possibly understand Byzantine society without firmly understanding the importance of their religion and its theology. If you find yourself interested in the army of the late 11th century and you can find a copy of his book The Development of the Komnenian Army, I would recommend it. He didn't use it in our course, but I got access to a copy later on and very much enjoyed it.

Best of luck in your search!