Paul quotes two Greek sources here as support for his point that the creator God does not need temples or service from humans. The first allusion is to Epimenides the Cretan, a poet also cited in Titus 1:12. The original poem no longer exists, but it appears in a number of other ancient writers. The second citation is from Aratus, a Cilcian poet (Phaenomena 5). The original line, “in him we move and live and have our being,” was pantheistic, but Paul spins this line into a statement about God as the source of our life.
We might ask how Paul came to know these lines of poetry. There are not many modern readers who can quote freely from current poets or philosophers. One possibility is that he had some secular education which could be applied to the preaching of the gospel. We might imagine Paul thinking through his task of being a light to the Gentiles and researching possible points of contact in order to preach to pagan audiences. This is in fact a typical way of doing apologetics today. Christians will study philosophy for the purpose of interacting with the philosophical world in their own terms. While I do not think this is a bad idea at all, that may not be Paul’s point in using these sources.
On the other hand, these may very well have been well known bits of proverbial wisdom that were more or less “common knowledge.” If so, then the allusion to Greek poets is more like the preacher who uses a common phrase in order to make his point. Or better, Paul is quoting lyrics of popular songs to make his point. I occasionally use a line from a popular movie or song in order to make a point (although with my taste in music, it usually does not work very well.) This comes down to knowing your audience. I have found that I can get a lot further with college age group with a Simpsons reference, while the same line is lost on an older adult group. Perhaps that is what Paul is doing here in Acts 17 – he is riffing on the culture.
(Let me comment here that most of the books which try to use movies to teach the gospel with a popular movie are lame and probably only read by Christians who like the movie in the first place. I cannot imagine that any pagan would pick up “Finding Jesus with Frodo” and get saved as a result.)
In both of the allusions Paul simply intends to show that his thinking is not all that far from authorities which the audience would have understood and appreciated. To cite the Hebrew Bible would have been fruitless since the audience did not know it, nor where they well disposed to hearing from Jewish texts! Paul does not think that Jewish or Christian theology can be added to Stoicism in order to put one right with God – there must be a conversion to an entirely new worldview.
Does this mean that Acts 17 is permission to quote The Simpsons or Bob Dylan in sermons and Bible studies? Perhaps, but we need to couple cultural reference with a serious point from the text of the Bible. It is one thing to mimic culture to attract attention to you point, but it is a fairly worthless strategy is if there is no point behind the reference. I think that you can (and should) illustrate serious theological points via cultural artifacts (like poets, books, movies, etc.)
If the point is obscured by the fact that you rolled a Family Guy clip in church, then you have missed Paul’s point.