I suppose a general note would be in order, then, before beginning: When we do use the OT, we use it quite extensively. According to the letter of our Liturgical guidelines, we cover the whole Psalter every week, and during Great lent we read through most of Genesis, Proverbs, and Isaiah. Add that to extensive pericopes attached to the various feasts of the Lord, His Mother, and the Great Saints, and we do indeed hear quite a bit of the OT.
However, generally the OT is the preamble, the build-up, the shadow of the salvific work of Christ. With the coming of the Lord it (the OT) has not passed to obscurity, but its interpretation has now been radically altered by our interaction with the incarnate Son of God. The Divine Liturgy focuses on the Kingdom of God, and thus utilizes the writings about Christ's earthly ministry and the stories and exhortations of those who lived in His renewed Church (excepting our use of a few doxological psalms in the beginning). The Matins is a bridge between the Old and the New, infusing the story of humanity with the new (from our POV, not God's) reality brought forth by the Incarnation, Death, and Resurrection of Christ - so the OT is infused in the hymns (and the readings from the Psalter), but the readings (when present) are from the Gospel.
In the Vespers, the recounting of the creation of the world (especially in the Proemial Psalm that "leads off") makes the Vespers an ideal time to bring forward the stories and teachings of the OT which have found their fulfillment in the life and teachings and transformation of the Lord. Here, they fully have their home: readings from the Psalter, and from various other books of the OT, pepper the celebration of Vespers, which is the beginning of the Liturgical Day and, thus, the beginning of our meditation on any feast or saint.
This, of course, should not discount the fact that our divine services are dripping from every paragraph, sentence, and phrase with the divine scriptures, both OT and NT. You cannot travel even 5 words without encountering a quote or reference to the scripture in our Liturgical texts, even (or especially, depending on your POV) in the prayers and exhortations of the clergy (petitions and exclamations, and everything in between).
Now this seems a bit muted during the year, since most Saints do not have OT readings assigned to their celebration. This is a reflection of human reality, not divine: the Church has chosen to prioritize the celebration of feasts and saints for the benefit of, and reflecting the piety of, the people. St. George isn't inherently a better Saint, or closer to God, than St. Silouan, but (a) people have felt more of a connection with him, and (b) God has chosen to work wonders through him. So St. George's feastday features Vesperal OT readings, and St. Silouan's doesn't, but this prioritization doesn't reflect on the "inherent" holiness of either one.
Personally, I would counsel someone to read the NT thoroughly, many times, before going back over the OT. We cannot choose to interpret the OT in a vacuum, since the Incarnation, Death, Resurrection, and Ascension of Christ radically changed all creation - we must proceed then reading the OT using that revelation and reality as our lens.