The journey of memorizing a whole book of the Bible begins with memorizing a single verse, to borrow a phrase. And so begins the journey. After much preparation, now begins the true substance of making progress toward hiding God's Word in his heart.
One may have already memorized a verse of the Bible, so this is going to be mostly introduction for those less familiar the process. Having said that, some veterans of Scripture memory may be surprised at the small number of techniques recommended.
My first step is to simply read the verse out loud, about three times. Very easy. The third time may already begin to feel like you don't need to read it anymore, and that's a good sign. If that doesn't happen right away, and one wants to keep reading until that happens, that works, too. This doesn't even feel like memory effort is yet required, and yet memory things are already happening! One is making sure he knows how to pronounce all the words. A sense of the rhythm of the text will begin to form. Look at the grammatical structure and see how the parts of the verse fit together. Just like some consider rest the first stage of sleep, so reading is the first stage of memorizing. Average time: 1-2 minutes.
Second, look again at the first phrase or thought and then try and say it without looking. If possible, double-check it right away, and then say it three times without looking again. Average time: 1 minute.
The next step has a couple options to it. One can either repeat step 2 with the next phrase by itself, or repeat step 2 with adding on the next phrase to the first. Adding phrases cumulatively can take more time at first, and taking another phrase separately may require more of what I call “stitching” together later. Whichever is comfortable or works well at the time is acceptable. Either way it all gets put together very soon anyway. As one makes progress through a verse or puts the learned parts of the verse together, one may find early getting-stuck points in how well you know what you just learned. This is very normal and nothing to be discouraged about. It's just a reminder to go back through the verse and keep practicing what you learned until it is down smoothly. (If one finds he is continually getting stuck on one part, it's good to pause work on the whole verse and focus on just a small problematic part of it until the challenges get worked out.) Repeat this process until one learns the whole verse. Note: verses can vary in length, so at times one may prefer to take on more or less than a single verse. That is entirely acceptable. The placement of verse numbers is not inspired anyway. How much one memorizes at a time is not important, but rather that one is memorizing the words themselves, how ever many that may be in one sitting. Average time: 5-10 minutes.
Finally, once the verse is memorized, reinforce it. For me this means reciting what I have learned out loud three times toward each direction of the compass. Three times in four directions for a total of 12 times. This is an important step because one's brain also associates what one has learned with what one sees as he was learning it. It is important to break the visual association in order to know the words no matter where one is looking. It's surprising how rotating 90 degrees can throw off one's memory of what one just learned, and yet this shows this step is necessary and helpful. To avoid the distraction of thinking about the count for each direction while thinking about words, I keep the count on my fingers. Average time: 3-5 minutes for reciting a single verse 12 times.
Total average time: About 20 minutes, maybe less, to learn a verse and have it locked away near rock solid. Set aside some time each day, and this can be life-transforming.
There are a variety of other memory techniques people have tried and use for memorizing short passages. Considering the purpose here of memorizing long passages of Scripture, I don't recommend them, and the few that may be necessary I only as a last resort. Putting the words to song would make the already-long 36 minutes it takes to quote the entire book of Romans exponentially longer. Mnemonic devices can also be more distracting from the meaning than helpful for memorizing. Building an acrostic to remember key words is a lot of extra work. Few of these external methods emanate from the text itself or are built on the meaning of the text.
I recommend simple repetition, hearing the words, thinking about their meaning, using the natural structures conveyed in the text and in the meaning of the text, getting a sense of its natural rhythm and pacing, more repetition, and practice in a variety of places for learning and making sure one remembers the words.