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Where to begin
Don't memorize randomly. Start with passages that matter to you—ones you return to, ones you need. Meaning sticks better than abstraction.
Below are passages I recommend for those beginning a memorization practice. They're manageable in length, rich in doctrine or comfort, and they're often ones you'll want to recall.
Suggested starting passages
John 1:1-5
The logos, light, and identity of Christ
Romans 8:28-39
Assurance and God's faithfulness
Psalm 23
Comfort, provision, guidance
Philippians 4:4-7
Joy, peace, prayer
2 Timothy 2:1-3
Endurance and grace