Don't set out to read one a day, or some such thing, because any given psalm may offer you very little. You may not suffer because of enemies, and a surprising number of psalms concern that. You may not have any complaints against God, and many psalms give voice to those. It is better to read down through the poems at random, taking a chance that in some couplet or image you'll suddenly hear The Bible speaking for you. People justly love the Psalms for at least two reasons.
Most psalms come across as less preachy and more personal than much devotional writing.
Another is that the writer whose heart is exposed to ours through his or her psalm very often suffers in a way familiar to us, or wonders in a way we recognize. People who feel out of place reading through the Sermon on The Mount may turn to the psalms and find plenty of company in their self-pity, persecution, pleading, and talking oneself into being brave. They are not theologians' craft, nor saint's confession, but human nature. And yet there is profound theology in it, and rich resource for the inspired heart. Some psalms have the grandeur of cathedrals, and some the power of great music.