How does one testify to the Orthodox faith as being the only correct way of worshipping God?
When telling others of the Orthodox faith, I suggest you "share it" with them instead of telling them of it. Sit in a non-threatening environment and look at the New Testament together, specifically the Holy Book of Acts and the birth of the Church at Pentecost. On that day, the Holy Spirit descended on the Twelve Apostles and those gathered in the Upper Room and by afternoon some three thousand souls had believed in the Lord Jesus Christ and were baptized due to their faith. The Holy Scriptures further tell that when the very first Christian congregation began, "they continued steadfastly in the apostles' doctrine and fellowship in the breaking of bread, and in prayers" (Acts 2:42).
It is not difficult to trace the early Church history up to the present. Many books record the history of the Orthodox Church to present. Books such as: "2000 Years of Coptic Christianity" (Meinardus), "Introduction to the Coptic Orthodox Church" (Fr. Tadros Malaty), and "Traditional Egyptian Christianity" (Patrick) are very good referral books both for you and someone interested in the history of the Coptic Church. "Coming Home" (Fr Gillquist) and "Becoming Orthodox" (Fr Gillquist) are both interesting reading for those outside the faith looking into Orthodoxy.
If we are to evangelize to others, our aim should be to win them to the Orthodox faith and not simply for the sake of criticizing, and finding fault in another faith. Let your focus be on your own faith, what is good and true about its apostolic foundation and steadfastness. It is enough that over all these centuries, Orthodoxy has kept the same faith delivered to the apostles and saints.
Always extend to everyone the opportunity to attend the Divine Liturgy and examine the Orthodox faith and worship. Through your example, love, and teaching church evangelism can become an ongoing process.