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.
Book One:
Psalm 7: A prayer for deliverance from enemies. A lament.
Superscription: We have no story of a Benjaminite named Cush. This psalm may be in response to an event not recorded in the Bible. “Shiggaion” is a musical term which only appears this one time. Scholars do not know what it means.
Verses 1-5 An innocent person could take refuge in the sanctuary. These verses include a cry for help and the statement of innocence. Verses 6-9 ask for God to judge, recalling God’s status a judge of the peoples. The assembly may be the heavenly court. Verses 10-11 describe God as judge and righteous and savior. Verse 12-16 concern the fate of the unrepentant. Notice here that people destroy themselves by their own actions. Verse 17 concludes the psalm with thanksgiving for God’s actions.
Psalm 8: Wonder at God’s Creation of Humans: A Hymn of Praise. “The Gittith” is a musical term of unknown meaning.  Notice that verses 1 and 9 are the same. Also notice how both human specialness and lowliness are both part of this psalm.  In verse 1 “name” is paralleled by “glory”. In the introduction to the Psalms we mentioned that Hebrew poetry used parallelism. Sometimes one line reinforces or amplifies another, sometimes a contrast is drawn. Where do you see parallelism in this psalm?
Psalm 9-10 : A lament, a prayer for deliverance from enemies. In the Hebrew bible psalm 9 and 10 are two psalms. In the Septuagint, these two Psalms are a single Psalm and they form an acrostic,  each section begins with a letter of the Hebrew alphabet. Psalm 10 does not have a title, as other psalms do, which suggests 9 and 10 belong as one psalm. Because this psalm is an acrostic, there is not much in the way of a logical sequence. It moves between hymn, supplication, reflection, thanksgiving and confidence.
“Muth-labben” is another musical term whose meaning is unknown.
9:1-2 A promise to tell what God has done
9:3-4  tell of what God has done for the psalmist
9:5-8 what God has done for the nation
9:13-14 a cry for help during an attack
“Higgaion” a musical term of unknown meaning.
10:1-18 express a desire for justice and God’s intervention.  The psalmist asks “why’?  Verses 3-11 concern the wicked and the suffering of the righteous. Verses 12-18 call to God for action.
Psalm 11: a song of trust. Confidence in God’s concern for justice.Verses 1-3 the psalmist rebukes those who call on the people to flee their enemies. Verses 4-7 The palmist calls for people to trust God. The wicked will fail and the righteous will be saved.

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.