Following my enthralment to the TV series Game of Thrones (and this is not the second series which is playing as of now in the US) I began, as soon as I could get my hands on Part One, reading A Song of Ice and Fire, the books that the TV series is based on.
It is probably possible to read a book right through. I haven't found it the best way for me. With twelve or fifteen viewpoint characters, I quickly started leafing through the book and getting continuity of each character in turn, or each couple of characters. For example, it was fairly easy and propitious to read Sansa's story and Tyrion's at the same time in Part Three, first half.
There's no problem yet in reading Daenerys's story separate from the others, it doesn't intersect very much yet with any of the others. The same with Jon Snow's adventures, though to get the fine detail on the interaction of the Stark's dire wolves, I also had to have Bran's part handy.
After I did the same thing with the first half of Part Three, I had to go back to Part Two for more detail. Lucky I still had them both on hand. There's a long line of reserves at the local library to wait through before I can get the second half of Part Three.
I wondered about the increasing size of the books. I remember that blow-out happening with the Harry Potter books as well.
However, I discovered with pleasure that Martin is an exquisitely courteous writer. Sol Stein, in one of his how-to books, named courtesy as a necessity in the relationship authors seek to have with their readers. One of the courtesies invoked by Martin is his practice of reminding readers of what went before. Essential in such a complex ongoing story as this one, with so many viewpoint characters. Take this example in Part Three of Catelyn talking with her son Robb, the King of the North (163):
"Any man Grey Wind mislikes is a man I do not want close to you. These wolves are more than wolves, Robb. You must know that. I think perhaps the gods sent them to us. Your father's gods, the old gods of the north. Five wolf pups, Robb, five for five Stark children."
"Six," said Robb. "There was a wolf for Jon as well. I found them, remember? I know how many there were and where they came from. I used to think the same as you, that the wolves were our guardians, our protectors, until ..."
This whole section says nothing much new, but says it all in a new way, largely by Catelyn, who in Part One didn't believe in the old gods and who felt like an outsider in the Winterfell godwood, giving readers another aspect of her character arc as well as reminding them of the mythology around the dire wolves.
Part Two and Three are full of that kind of reminders; that add to knowledge about the character under consideration, as well as serving as a place where to weave in new world building/background facts.
For me, this courtesy by Martin is a big part of what makes A Song of Ice and Fire such an amazing reading experience.