As you do Bible memorization for yourself or with your children, consider memorizing whole books or at least large sections of books.  The benefit of this approach is that you will know the context of verses to understand them correctly, spend less time on memorizing verse references and have a clear memorization plan.  You can also find audio Bibles that will read you the books to help with the memorization process.  The cost of tackling long passages is that you will need to spend time on a few less useful Bible verses that fall within the book.  The following books and passages provide a reasonable balance of highly concentrated material organized from shortest to longest:

• Psalm 1
• Psalm 100
• Hebrews 11
• Romans 5, 8, 12:1-13:14
• Matthew 5-7 (Sermon on the Mount)
• Colossians: concentrate on 1:9-4:6
• Philippians: can focus on 1:3-2:18, 3:7-4:9
• James: good book for kids to start with because there are fewer difficult theology concepts
• 1 Peter
• Ephesians: it’s difficult to skip any parts, but you could consider not learning 3:1-13
• Galatians: can focus down to 2:15-6:10
• 1 John 1:5-5:12

If you’re up for tackling on of the longer books first, start with Ephesians.  It has a nice balance of doctrine in the first three chapters followed by application in the last three.  It teaches about the nature of grace, the sacrifice of Christ and the mystery of the Gospel along with unity in the church, putting off the old self to imitate God, relationships and the armor of God.

There are also several good lists for additional ideas depending on your memorization goals:

• Most popular verses, chapters and books
• Navigators list of top 40 chapters, which concentrate on Genesis, Isaiah, Matthew, John, Romans, Ephesians and James
• Key chapters by topic such as love, humility, salvation and suffering and another view of this
• Crossway’s view of the most popular verses

There’s also an article that supplements this one with a more prioritized Bible verse list