They speaks to the full range of human emotion and are rightly identified as giving permission to say what you're actually feeling. A Christian finds themselves inside the conversation of the Father, Son and Holy Spirit and needn't worry about what's appropriate to say - there is room to rage as well as to adore.
Six things I find helpful in navigating the Psalms.
#1 Five books.
There are five books of Psalms, marked out in the structure in most Bible's. There is a rough correspondance in themes between these and the five books of the Pentateuch. You'll find lots about trees and the blessed man in Book 1, as in Genesis. Deliverance in Book 2, like the Exodus. The Sanctuary in Book 3 etc.
#2 Headings matter
Don't skip the bits about David or the Sons of Korah, this sets the context. The Sons of Korah are resurrection men (check the background in Numbers). Don't miss Maskil, Gittith and words like Selah.
#3 Psalms on the lips of Jesus
Before you put a Psalm in your mouth, hear it in the mouth of Jesus. Jesus prays these before and more fully than we ever do. Steve Collier shows how to do this beautifully here: Psalm 22, Psalm 23 and Psalm 24. When you're with David in the Bible its always worth asking, how would Great David's Greater Son pray this Psalm?
#4 Read it like a book
Psalms feels like a song book where we can assume that you can just dip into Number 4 and then number 78 and then number 114. And you can, but the songs are arranged purposefully, not by Title or Theme but as sections of Scripture next to one another. Some of the plot is particularly clear (as above in Psalms 22,23,24).
#5 Let the Psalms speak of Jesus
These are songs of The Blessed Man, set on the Holy Hill whom the world raged against - take refuge in him. Hear the Son cry to his Father. Hear the gospel loud and clear in the Psalms like the NT writers, early church and most Christians until the last couple of centuries have. The second Adam rejoices and exults, and in his death knows what it is to be forsaken and to thirst. Songs of the Exodus and the Sanctuary and of persecution in the wilderness all speak of Jesus. We always need to "hook on" to the bigger story. Let the Bible's story of the Triune LORD, of Jesus the LORD who saves fill the Psalms with meaning.
Iain Campbell: "if Jesus is not the God of the Psalms, I do not know who he is at all."
In the end, sing. Join the songs of the Father, Son and Holy Spirit. Feel liberated to sing, and sing all the more richly as you let the Psalms sit in the context of their book and of the whole sweep of God's story.