The Psalms have been the prayer book of the Church since its founding. Jesus and the apostles were familiar with the Psalms as evidenced in them being one of the most quoted books throughout the ministry of the early Church. It is important that we learn to pray the Psalms well in order to be in a thriving relationship with God. Here are some varies ways to read the Psalms and reflect on them:
Read according to the date

If it's 4/13, drop the second number and read Psalm 43, that way, you will get through each Psalm you read about three times a month.

Read them in different translations

We live in a day and age in which there are dozens of faithful Bible translations. Read the Psalms from different translations each week, month, or year to develop your prayer vocabulary with God!

Read in your car before walking into work/school

It is so much easier to remain focused on God wherever you are when you read the prayer of someone pouring out their heart to Him.

Read in community

-Call and response with each verse
-Read the whole Psalm together

First time through the Psalm:

Read the entire Psalm

The best way to grasp the theme of each Psalm is to read it in its entirety (maybe with the exception of Psalm 119!)

Read a Psalm and then immediately pray to God after having read it, trying to use some of the language in your prayer.
Second time through the Psalm:

Read section by section

Oftentimes, Psalms will have natural breaks where the flow of thought or reference point changes. Use these to stop yourself and reflect more deeply on the section you just read and let the truth of it sink in before moving on.
Third time through the Psalm:

Read verse by verse

See how this verse adds significance to those around it. Ask, how would the Psalm be different if this verse were not in it?
Fourth time through the Psalm:

Read sentence by sentence.

Let the truths of each sentence sink deep into your heart. If you can't genuinely pray the sentence to God, think of a time when you could or a believer who could at this point (i.e. someone who is being persecuted).
Fifth time through the Psalm:

Read each verse multiple times in a row, emphasizing different words.

How does the emphasis add significance to the truths in this Psalm and shed light on God's Word?

Example:
1st time through: The Lord is my shepherd
(reflect)
2nd time through:
The Lord is my shepherd
(reflect)
3rd time through:
The Lord is my shepherd
(reflect)
4th time through:
The Lord is my shepherd
(reflect)