Fighter Verses is a great website to help you learn and actually do Bible memorization: http://www.fighterverses.com/. You can see the different sets of verses that they recommend you memorize at http://www.fighterverses.com/the-verses/. while I recommend to memorize passages or books of the Bible, this is good for starting if you would rather do it this way. On their website they describe their Bible Memory Plan in this way:
The Fighter Verse Scripture memory plan is built on and powered by the belief that Jesus and his gospel are “the power of God for salvation to everyone who believes” (Romans 1:16)—not only the power for the initial faith of conversion, but the power for the everyday Christian walk of faith. The Fighter Verses have been carefully selected—52 passages per year for five years (260 total)—to arm us for the fight for faith and love, and against unbelief and sin. While the whole Bible is indeed both about Jesus and “profitable for teaching, for reproof, for correction, and for training in righteousness,” our team has chosen the passages that we sensed would be most helpful in fighting for faith in God, his Son, and his gospel, as well as against particular sins and unbelief. The passages fall into three major categories: 1) those that fix our attention on God, his power, and his compassion toward us who believe; 2) those that arm us for the battle against specific sins; and 3) those that turn our attention to the person and gospel work of Jesus where the strength for faith, and defeating sin, most powerfully flows.
Click on the link to see some resources to help you start to memorize the scripture: http://www.fighterverses.com/memory-resources/. Some of the Great Resources include: 6X6 Bible Memory Sheet, Bible memory Progress Chart, Bible Memory Chart – Sunday School, An Extended approach to the Memorization of Scripture, Bible Verse Card Maker