Michael A. G. Haykin gives six reasons that we should read and study the church fathers (Rediscovering the Church Fathers: Who They Were and How They Shaped the Church [Wheaton: Crossway, 2011], 17–28):

1. For freedom: “[S]tudy of the Fathers, like any historical study, liberates us from the present.”
2. For wisdom: “[T]he Fathers can provide us with a map for the Christian life.”
3. To understand the New Testament: “We have had too disparaging a view of Patristic exegesis and have come close to considering the exposition of the Fathers as a consistent failure to understand the New Testament.”
4. Because of bad press about the Fathers: “[T]hey are sometimes subjected to simply bad history or bad press.”
5. As an aid in defending the faith: “The early centuries of the church saw Christianity threatened by a number of theological heresies: Gnosticism, Arianism, and Pelagianism, to name but three. While history never repeats itself exactly, the essence of many of these heresies has reappeared from time to time in the long history of Christianity.”
6. For spiritual nurture: “The study of the church fathers, like the study of church history in general, informs Christians about their predecessors in the faith, those who have helped shape their Christian communities and thus make them what they are. Such study builds humility and modesty into the warp and woof of the Christian life and as such can exercise a deeply sanctifying influence.”

So where should you start? Haykin suggests some books in Appendix 1 (157–58, bullet-points added):

Reading the Fathers: A Beginner’s Guide

[Introduction]

Where does one begin reading the Fathers? Well, first of all, I would start with two tremendous secondary sources:

• Robert Louis Wilken, The Spirit of Early Christian Thought: Seeking the Face of God (Yale University Press, 2003) and
• Henry Chadwick, The Early Church (Penguin, 1993).

Together these will provide an excellent orientation in terms of the history of the Patristic era (Chadwick) and the spirituality of the Fathers (Wilken).

• If one is so inclined, Jaroslav Pelikan’s The Christian Tradition: A History of the Development of Doctrine, vol. 1, The Emergence of the Catholic Tradition (100–600) (University of Chicago Press, 1971) is the finest introduction to the thought of the Fathers. While not an easy book, it is a gem. [Haykin reflects on this book in Appendix 2, pp. 159–65.]

[Surveys]

• For a good overview of the period, see the relevant pages in Tim Dowley, ed., Introduction to the History of Christianity (Fortress, 1995),
• and for the key leaders, see the biographies in John D. Woodbridge, ed., Great Leaders of the Christian Church (Moody Press, 1988). The latter is regrettably out of print, but secondhand copies can be gotten easily.
• I have also had published Defence of the Truth: Contending for the Truth Yesterday and Today (Evangelical Press, 2004), which deals with theological challenges faced by the ancient church.

[Primary Sources]

Of course, one must not avoid getting into the Fathers directly. Any advice here is bound to be somewhat eclectic, but

• I would recommend starting with Augustine’s Confessions , the masterpiece of Patristic piety.
• Then I would read, not surprisingly, Basil of Caesarea, On the Holy Spirit, which, as we have seen, is a masterly combination of fourth-century piety and theology.
• The second century Letter to Diognetus is an excellent entry point into early Christian apologetics, and
• The Odes of Solomon an overlooked gem of worship, also from the second century.
• In recent years I have had a renewed interest in the Latin tradition, and here I would recommend Cyprian’s Letter to Donatus and
• Hilary’s On the Trinity, book 1, which recounts the story of his conversion.
• In the Patristic era many were impacted by Athanasius’s Life of Antony . Personally I find this work somewhat off-putting even if it is a fascinating window into early monastic thought.
• I much prefer Gregory of Nyssa’s warm account of his sister, The Life of Macrina .
• Finally, Patrick’s Confession is a must read for reasons enumerated in chapter 7.

Perhaps the best place to start is with Haykin’s readable book.