Let me give you a little insight into how I have started to memorize Scripture:
1. Choose a passage. Start small or find something that you are "fairly" familiar with but do not have memorized. The reason I can start small is that I am going to be in this for the long haul. I am not trying to memorize the whole Bible (or a whole chapter) my first week. But, over the course of several weeks, the small will accumulate into the large.
2. Write the passage on a 3x5 card or something that you will carry with you throughout the day. I have a thin Moleskine that I use for prayer requests, and I have started copying the Scripture passages starting from the back.
3. Read through the passage 4 or 5 times at the start.
4. Throughout the day, say the passage from "memory" referring to the 3x5 card as needed. I do this while I am driving (hey, people talk on their cell phones, I memorize Scripture), while I am waiting in line somewhere, while I am showering, etc. I do this at least 5-6 times throughout the day.
5. I look for ways to apply the verse throughout the day. In The Great Omission, Dallas Willard emphasizes our role in the spiritual growth process. As we submit ourselves to the spiritual disciplines and practices of the Church God gives grace. If we do not participate in these disciplines and practices, we find that we are not in a position to receive God's grace. I find that memorizing the Scripture has led to my recognition of the application and appropriateness of God's word to my life throughout the day. Every time I see a place for application, I say the passage again.
I am tempted to read large passages of the Bible because all Christians are supposed to read large portions of the Bible right? But, not to discourage this practice, I think I have come to the conclusion that it is better to "master" a small portion than to read large chunks and never "get it." I do think we need to read large portions of Scripture, meditate on them, and study them in such a way that does not remove the individual verses and passages from their overall context. But not at the expense of "mastering" at least some portion.
Reading large passages of Scripture does several things: it gives us a sense of context, it lets us see the thought process of the author and the Author, and we are immersed in the overall message. Memorization helps us master a passage, take the passage with us into life, and meditate on God's word. Both have their place, but not one at the expense of the other.