I completely agree with Andrew Davis’s recommendation that we work toward memorizing whole books of the Bible. From Joe Carter’s interview with Davis at the Gospel Coalition:
The Word of God comes to us, for the most part, in paragraphs—developed trains of thought that tell a story (narrative, Gospels), unfold a law (Pentateuch), make a doctrinal argument (epistles), make a case against God's people and their sins or reveal the future of God's people (prophets), or describe in complex imagery an apocalyptic vision of the future (Revelation). Individual verses do not capture the train of thought and therefore are more likely to be taken out of context.
The fuller the section of Scripture we can memorize, the better. Also, Jesus said, "Man does not live on bread alone but on every word that comes from the mouth of God" (Matt. 4:4). Memorizing individual verses usually gravitates to the more "famous" verses and likely misses some precious truths hidden in less familiar verses. Memorizing whole books gives people a sense of the perfection of every word of God. It is a continual learning experience, a journey of discovery in which details come alive with incredible power. I am in awe of the majesty of every word of Scripture, and that has come in great part from 30 years of memorizing less famous verses and seeing the wisdom of God in every letter.
I haven’t read his book yet, but it sounds like a good place for you to start, and eventually you’ll figure out the method that works best for you (e.g., I’ve found that memorizing while I’m walking increases my retention exponentially). Sometimes all it takes to get started memorizing a book of the Bible is just knowing that such a thing is possible.
It’s worth your time and effort. For me, reading a book of the Bible James Gray style for a month, coupled with extensive memorization in that book, has been life-changing. (I alternate every other month with reading straight through the Bible, picking up where I left off and reading as far as I can, so that I continue to get a regular dose of the big picture, as well.)