You will find an introduction to the Psalms here.
Here is a prayer to use before reading from Gregory of Nazianus (329-389), an early Church father.
Lord, as I read the psalms let me hear you singing. As I read your words, let me hear you speaking. As I reflect on each page, let me see your image. And as I seek to put your precepts into practice, let my heart be filled with joy. Amen.
Psalm 37: A wisdom psalm This is an acrostic psalm and encourages people to trust in the Lord and not “fret”. What things are the reader encouraged to do? What will happen to the wicked? What will God do?
Psalm 38:A lament. This psalm, in the Christian tradition, is the third of the seven penitential psalms. Remember in the ancient world, illness was believed to be a punishment for sin. Notice that in this psalm the psalmist does not claim innocence. In vivid language the psalmist’s situation is described both physically, emotionally, and socially. The psalmist, even though having sinned, still waits and trusts in God.
Psalm 39: A lament. Two other psalms are attributed to Jeduthun – 62 and 77. The psalmist has not spoken publicly about their illness until now so as not to give encouragement to the wicked. In verses 4-6 knowing that all human life is short, the psalmist asks how long he will suffer. The psalm ends with a prayer for healing.The last verse recalls Job 7:19 and asks that God not look too closely at him because if God looks closely, God will see his sin.
Psalm 40: A psalm of thanksgiving and petition. Verses 1-10 are a thanksgiving and 12-17 a lament. These may have originally been two independent psalms. What does the psalmist say is the appropriate response to God’s actions? Verses 12-17 ask for God’s continued help.
Psalm 41: A lament. Verses 1-3 are in the style of a wisdom psalm. Notice again the connection between sin and illness. Not only are the psalmist’s enemies glad his is ill, but his “bosom friend” has turned away from him. The psalm ends with a declaration of confidence in God.
41:13 is not part of the psalm but is a doxology that marks the end of this portion (book one) of the psalter.
Here are several good sources to aid your reading of the Psalms
Anderson, Bernhard W., Katheryn Pfister Darr, Understanding the Old Testament Abridged fourth Edition. (Upper Saddle River,New Jersey: Prentice Hall) 1998.
Clifford, Richard, “Psalms” in The New Oxford Annotated Bible with Apocrypha: New Revised Standard Version.Coogan, Michael D.; Brettler, Marc Z.; Perkins, Pheme; Newsom, Carol A. (2010-01-20) Oxford University Press. Kindle Edition.
Stuhlmueller, Carroll, “Psalms” in HarperCollins Bible Commentary Mays, James L. ed.(San Francisco: HarperSanFrancisco) 2000.
McCann,Jr, J. Clifton “Psalms”,in The New Interpreter’s Bible Vol 3, Keck, Leander E. ed. (Nashville: Abingdon Press) 1996.