You will find an introduction to the Psalms, here. Here is a prayer from Gregory of Nazianus (329-389) who was 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.
Book III Psalms 73-83 are psalms of Asaph. Psalms 84-85,87-88 are from the guild of Korah. Psalm 89 is from the psalms for Davidic kings.
Psalm 73 This is a wisdom psalm and is concerned with the question of why, if God is just, do the wicked prosper.The psalm moves from the psalmist’s concern, even despair to seeking answers to realization of the goodness of God. Verse 1 is a statement of faith. But the psalm continues to describe what the psalmist sees in the world and how the psalmist was tempted to respond. Verses 21-22 recognize that the psalmist’s previous attitude was foolish and the psalm ends with a recognition that God is present, near and all the psalmist needs.
Psalm 74 A community lament This psalm describes the destruction of the Temple and is a cry for help (verses 1-11). Verses 12-17 are a prayer of remembrance of God’s deeds. The psalm ends with a prayer asking God to remember God’s people.
Psalm 75 A psalm of thanksgiving. Verse 1 is a community expression of thanks. Verses 2-5 are an oracle from God. Verses 6-8 A warning. Verses 9-10 an individual ( perhaps the king) concludes with praise. Verse 4 “lift up a horn” is a way of talking about pride, arrogance and victory.
Psalm 76 A song of Zion celebrating God’s victory. This may be an early psalm because it refers to Jerusalem as the capital of Judah and Israel. This dates to Solomon’s time or earlier. Verses 1-3 are the announcement of God’s victory. Verses 4-9 describe the victory in a hymn. Verse 10 seems to mean that evil intentions or anger can be turned to praise of God. Verses 11-12 The congregation is encouraged to worship God.
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.