Paul's sermon in Acts, along with Stephen's sermon, differs a little from the other sermons. It still has the signature climax--God raised Jesus from the dead (17:31).
But it is quite possibly a good example of incarnational preaching. Paul meets the Athenians on their terms. He talks natural theology, about stars and the heavens. He quotes Stoic poets. He sees an altar to an unknown god and uses it to argue for the one true God.
Some possible insights here on incarnational ministry, especially to post-Christian environments!
Some clever beasts have tried to put this incident together with 1 Corinthians where Paul says he determined to preach only Christ crucified. This is ingenious but of course Paul never says anything of this sort: 1) Acts never suggests that Paul felt his sermon at Athens was a failure, 2) 1 Corinthians never says that he switched his focus after Athens.
Given the standards of history writing at the time, we cannot even be sure that Acts 17 gives us exactly how Paul approached his hearing. It could just as easily be Luke (as Thucydides the Greek historian did) giving us the kind of sermon a Paul might have preached on this sort of occasion. If so, this would not be an error, by the way, because it hit the target Luke (and God) had in view.
This appears to be a legal hearing. The Areopagus was where the city of Athens met at law (at least at one point), so it is quite possible that Paul is in some sort of preliminary hearing--in legal trouble again. There were plenty of gods to believe in at the time. Creating new ones--new cults especially--simply wasn't smiled on. Of course Paul's God was the God of the Jews, a known quantity at the time.
Paul apparently doesn't get much traction in Athens, a fact that is brought up repeatedly to argue against education. The common narrative is that education makes you lose faith, so don't go to college/seminary/etc. All I can say is that Christianity is in trouble if learning makes you lose faith. Wouldn't the most natural implication of that line of thinking be that Christianity isn't true, and that you have to be ignorant to have faith?
There are places that are only interested in listening to new ideas and don't go anywhere with them or do anything with them. That certainly isn't a Christian university, where the exploration of truth is done in the context of faith--faith seeking understanding. It is also not the university that trains a person to be able to get a job.
The impression of the sermon is that it has been possible, although perhaps not real likely, for people to find God through natural theology. But now God was bringing special revelation directly to them. God was now calling on the whole world to repent, Jew and Gentile.