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 22: A Lament. “According to the Deer of the Dawn” may be the musical tune the psalm was sung to. Verses 1-21 are the cry for help. Within these verses are seven sections. vv 1-2 the introductory cry for help. vv 3-5 a confession of faith. vv6-8 a lament. vv 9-10 prayer of confidence. vv11prayer. vv12-18 lament. vv19-21 prayer. Notice that vv 1-2 and 19-21 repeat the words “distant”, “save” or “salvation” and “help” and so form an inclusio.
Verses 22-26 express gratitude and verses 27-31 express trust in God’s sovereignty. Christians will recognize the first line of the psalm as having been quoted by Jesus from the cross. Notice how other parts of the psalm are also part of the gospels (See Matt 27:41-43, 46; Mark 15:34; John 19:23-25.
Psalm 23: A psalm of trust. This psalm is very familiar to most readers. In this psalm God is portrayed as shepherd and as host of a banquet. God is also referred to in second and third person (you and he). In the Ancient Near East, shepherd was an image used to describe kings and the king’s care for his people.
Psalm 24: An enthronement hymn. probably used as an entrance liturgy as the Ark was processed. Verses 1-2 proclaim the cosmic victory of God. Verses 3-6 describe who may enter and welcome God. Notice that the criteria have to do with moral living. Verses 7-10 are a call to welcome God the king.
Psalm 25: An individual lament. This psalm is an acrostic which explains why the psalm does not have a logical structure. Even so many of the parts of a lament are present. vv 1-3 the cry for help, vv 18-19 the psalmist’s situation, vv 21 the declaration of innocence. vv 8-15 the expression of trust, and vv 16-20 a prayer for vindication. In verses 1-7 and 15-22 the psalmist refers to self (I) and to God (you). in the middle verses God is addressed in the third person (he).
Some commentators recognize a chiastic structure to this psalm
  v 1, 22 call upon God
v2-3, 19-21 saved from foe and not shamed
vv4-5, 16-18 following the path of God and relief from afflictions and troubles
vv 6-7 , 11-15 covenantal love
vv8-10 the center of the psalm the covenantal love of God who leads the faithful
Psalm 26 A lament for deliverance from enemies. The reference to the alter in v 6 may mean the psalmist was a priest. Notice the psalm begins and ends with a reference to walking. The language of “hate” is a way of saying the psalmist rejects the wicked and stands in contrast with the declaration in v 8 of love for God’s house. Again the parts of a lament are present, vv 1-3 the cry for help, vv 4-7 the claim of innocence v 8-11 prayer for help.

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.