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 17: An individual lament. Verse 1 -5 is an appeal to God and a recitation of the psalmist’s faithfulness. Verses 6-14 describe the psalmist’s current need. Notice that in verse 6 that the psalmist expects God to answer. Some commentators think that verses 13-14 are saying that all gifts come from God. Other commentators think those verses express a wish that those who are persecuting the psalmist receive what they have inflicted on others. What do you think? The psalm ends with an expression of confidence.
Psalm 18 : A royal thanksgiving. Some think this psalm may have been sung at David’s coronation. A similar psalm is found in 2 Sam 22. The psalm begins in verses 1- 19 with a description of the king’s rescue. Notice the cosmic nature of the battle and the victory. Verses 20-45 describe the kings reward and righteousness. David has faithfully kept the covenant and so has God. Verses 46-50 conclude the psalm with a hymn of praise and thanksgiving.
Psalm 19 : A wisdom psalm. God is creator of nature and giver of Torah. In other ancient religions the sun was considered a god. For Israel, God was the one who made and controlled the sun and other natural phenomena and nature glorifies God. Verse 1-6 concern natural phenomena and verses 7-13 praise God’s revelation in Torah. There are six synonyms for Torah in verses 7-9, what are they? The psalm ends with a prayer that this psalm be acceptable to God.
Psalm 20: A communal petition for victory. The order of the psalm follows the sequence of a service before a battle. Verses 1-5 are a prayer for victory. Between verses 5 and 6 some type of liturgical action may have taken place. Verses 6-8 are an assurance, perhaps given by a priest or prophet. The psalm ends with a prayer. Notice that victory is a result of God’s actions.
Psalm 21 : A royal thanksgiving, probably a companion to psalm 20. God is praised in verses 1-7 for answering the king’s prayer. Verses 8-12 are an expression of trust that God will protect God’s people. Verses 13 is a concluding prayer. The king’s military role was understood to be the implementation of God’s justice in the world.

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.