2 Samuel 22 
As you read this prayer, I suggest you read it as if the Messiah is praying instead of David.  You will undoubtedly see many passages that are Messianic.  Reading from that perspective will provide greater insight into both Jesus and David.  It is also interesting to note Psalm 18 is almost identical to 2 Samuel 22.
In Acts 2:29-36 Peter says, "David...being therefore a prophet, and knowing that God had sworn with an oath to him that he would set one of his descendants on his throne, he foresaw and spoke about the resurrection of the Christ, that he was not abandoned to Hades, nor did his flesh see corruption.  This Jesus God raised up, and of that we all are witnesses."  Realizing David was a prophet through whom God spoke of the Messiah's death, and resurrection gives a different perspective to many of his Psalms.

I suspect questions will arise as you read this as if Jesus were praying instead of David.  Because of the many references to save or salvation, one question might be "From what did Jesus need to be saved?"  He did not need to be saved in the way of sinful men for whom He came.  However, Jesus did need to be saved in the sense expressed in verse three which concludes "you saved me from violence" until the appointed time.  
A question about Jesus being trained for war and destroying His enemies might also come to mind.  1 John 3:8 says, "The reason the Son of God appeared was to destroy the works of the devil."  1 Corinthians 15:24-26 says, "Then comes the end, when he delivers the kingdom to God the Father after destroying every rule and every authority and power...the last enemy to be destroyed is death."  

2 Samuel 22:51 concludes with the LORD saying He shows steadfast love to his anointed, to David and his offspring forever.  The Hebrew translated as offspring is singular.  Paul makes a similar statement in Galatians 3:16, "Now the promises were made to Abraham and to his offspring.  It does not say, 'And to offsprings,' referring to many, but referring to the one, 'And to your offspring,' who is Christ."

Holy Spirit, please give us eyes to see Jesus.  We especially want to see Him as we read and study the Old Testament.  We desire to understand the fullness of our Savior and King.  Lord Jesus, thank You for humbling Yourself to become like us so we might become like You and our Father. 

Psalm 78:4-8 
Asaph instructs the people to listen to his teaching, but from these verses, his focus audience is the children.  He wants the children of Israel to know the mighty works of the God of Israel.  His desire is for them to know the One in whom they can set their hope by keeping His commands, unlike their fathers who were stubborn and rebellious.  

The LORD's stated purpose for choosing Abraham is relevant to Asaph's Psalm.  Genesis 18:19 says, "For I have chosen him, that he may command his children and his household after him to keep the way of the LORD by doing righteousness and justice, so that the LORD may bring to Abraham what he has promised him."  Moses reaffirmed this instruction from the LORD in Deuteronomy 6:6-7, "And these words that I command you today shall be on your heart.  You shall teach them diligently to your children, and shall talk of them when you sit in your house, and when you walk by the way, and when you lie down, and when you rise."
Father, stir our hearts to teach our children and grandchildren who You are and Your instruction for how we are to live.  Holy Spirit, please be actively involved in the relationship we have with those children the Father has placed in our care.