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 85: A group lament. Notice that this communal lament is not as intense as some other Psalms, ( see the Psalms of Asaph 74,80) Verses 1-3 are a hymn of confidence, recalling God’s past actions. Verses 4-7 are a prayer asking for God’s saving actions. Verses 9-13 describe what Divine forgiveness and faithfulness are like.
Psalm 86: An individual lament. The only psalm of David in Book III. Verses 1-7 are the cry for help. Verse 8-10 praise God. Verse 11 asks for guidance. Verses 12-13 are thanksgiving for what God will do. Verse 14 describes the particular need. Verses 15-17 are supplication.
Psalm 87: A song of Zion. This psalm may be from the sixth or fifth century BCE and function to reassure exiled Israel of their true home. Zion is another way of speaking about Jerusalem. Verse 3 provided the them for Augustine’s City of God, and also inspired a popular hymn. Verse 4 Rahab is another name for Egypt. All the cities mentioned in verse 4 are not part of Israel. Being born in a foreign land does not preclude one from being a citizen of Zion. Verse 7 appears to be a fragment or a citation of a song.
Psalm 88: A lament. One of the most desperate prayers in the psalter. “Heman the Ezrahite” was a wise man ( 1 Kings 4:31) or a temple musician (1 chr 6:33) The psalm has three sections, verses 1-9a; 9b-12; 13-18. Each section begins with a cry to God. Sheol and the pit are the underworld where the dead are. Abadon (v11) is another name for the place of the dead. The psalmist blames God for his problems. The psalmist asks God to consider if the dead can praise him.
Psalm 89: A communal petition. This psalm is part of the royal psalms and is the last psalm of book III. This psalm is quoted or alluded to in Peter’s Pentecost speech-verse 3-4 (Acts 2:30), in Paul’s speech at Pisidian Antioch – verse 20 (Acts 13:22), in Mary’s song -verse 10 (Luke 1:51) and in Zechariah’s song – v 52 (Luke 1:68).
This psalm asks God to honor God’s promise about Davidic kings. Notice the repeated words, -steadfast love, faithfulness. Verses 5-14 recall the creation of the universe and God’s supremacy. Verses 15-18 Just as God is supreme and praises in the world, so to God is praised by God’s people. Verses 19-37 are about the covenant with David and the Davidic dynasty. The ‘faithful one” is David or perhaps Nathan. Verses 38-51 A Davidic king has been defeated and the people ask God to remember the covenant. Verse 52 is not part of the psalm but is the doxology that ends Book III.
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.