Which psalms should you memorize? Memorize the ones that strike your heart as you are reading them. Each person will find different psalms to be more effective for himself. Begin with Have mercy on me, O God (Psalm 50); then Bless the Lord, O my soul (Psalm 102); and Praise the Lord, O my Soul (Psalm 145).
These latter two are the antiphon hymns in the Liturgy. There are also the psalms in the Canon for Divine Communion: The Lord is my shepherd (Psalm 22); The earth is the Lord’s, and the fulness thereof (Psalm 23); I believed, wherefore I spake (Psalm 115); and the first psalm of the evening vigil, O God, be attentive unto helping me (Psalm 69). There are the psalms of the hours, and the like. Read the Psalter and select.
A Prayer Rule, by St. Theophan the Recluse
http://agapienxristou.blogspot.ca/2012/10/which-psalms-should-i-memorize.html