Here’s what some of the apologists up through the end of the second century argued:
St. Justin Martyr (d. 165) tried to demonstrate that Greek philosophy was inferior to Christianity (Hortatory Address to the Greeks), and that the Christian worship of God was superior to pagan materialism and the pagan gods (First Apology, 9-10, 13, 25). St. Justin admits that on some points Christians and non-Christians agree, but on points where they disagree the Christians have “fuller and more divine” teachings (First Apology, 20). St. Justin says that in becoming Christians people abandon their immoral lifestyles (First Apology, 14-16, 28-30), that there is an afterlife (First Apology, 18-19), and that people should be afraid of going to punishment in hell (First Apology, 19; Second Apology, 9). Two other evidences that St. Justin gives are that the coming of Christ was foretold by the Hebrew prophets (First Apology, 31-53), and that the witness of the martyrs proves the truthfulness of their claims (Second Apology, 12).
Tatian (d. c. 173) argues that the non-Christian philosophers are unworthy of being followed (Address to the Greeks, 2-3, 21, 25), as are the heathen gods (Address to the Greeks, 10). He says that people are following demons (Address to the Greeks, 8, 12, 16-19) Tatian says that men fell from grace, and that only the Christian God offers a way to overcome death by having faith (Address to the Greeks, 15). Tatian also attacks heathen morality (Address to the Greeks, 22-24). He argues that Christian philosophy is more ancient than that of the Greeks (Address to the Greeks, 31, 36-41), and that it is better suited for all peoples and not just rich males (Address to the Greeks, 32-33).
St. Aristides the Philosopher (d. mid-2nd century) tries to demonstrate that the gods and philosophy of the Greeks is inferior, immoral, and ultimately not a world view that matches reality (Apology).
St. Theophilus of Antioch (d. late-2nd century) says that the Christian view of God is the correct one, though faith is required (To Autolycos, 2-8), and offers evidence for the truth of the resurrection (To Autolycos, 13). The pagan gods, on the other hand, are absurd and immoral (To Autolycos, 9-10). St. Theophilus also offers his own conversion as an encouragement to convert, and says that people who don’t become Christian are in danger of going to hell (To Autolycos, 14)
St. Athenagoras the Apologist (d. late-2nd century) argues that Christian monotheism is superior to pagan polytheism (A Plea For the Christians, 4-8, 17-23, 28-30) and materialism (A Plea For the Christians, 15-16). He mentions the testimony of the prophets briefly (A Plea For the Christians, 9), and also argues that the morality of Christians is far superior (A Plea For the Christians, 11, 32-35).
Once we get into the third century the works generally start getting longer, but all the arguments seem to be roughly the same. Not that I'm discounting later works, of course: surely texts like Against the Heathen by St. Athanasius are brilliant and insightful. It's just that, since this post is just giving an brief overview, I wouldn't be saying much more for St. Athanasius than I've said for the above writers. Anyway, I'm not sure that this is what you wanted, but it at least maybe gives you a sense of the types of arguments they used.