"Now while Paul waited for them at Athens, his spirit was provoked within him when he saw that the city was given over to idols. Therefore he reasoned in the synagogue with the Jews and with the Gentile worshipers, and in the marketplace daily with those who happened to be there. Then certain Epicurean and Stoic philosophers encountered him. And some said, 'What does this babbler want to say?' Others said, 'He seems to be a proclaimer of foreign gods,' because he preached to them Jesus and the resurrection. And they took him and brought him to the Areopagus, saying, 'May we know what this new doctrine is of which you speak? For you are bringing some strange things to our ears. Therefore we want to know what these things mean.' For all the Athenians and the foreigners who were there spent their time in nothing else but either to tell or to hear some new thing" (Acts 17:16-21).
Paul, whose heart was devoted to sowing the gospel seed and saving lost souls, could not sit around idly while waiting for Silas and Timothy to join him. He knew there was work to be done, particularly after seeing that Athens was full of idolatry. Paul, as was his tradition, taught those who gathered in the synagogue. Additionally, he shared the gospel message every day with those in the marketplace. Paul would reason from the Scriptures and preach about Jesus Christ to anyone who would listen to him! Essentially, Paul went to where the most number of people would be. This is a wise approach that any good "fisherman" will imitate.
Eventually, Paul was able to address some philosophers. Athens was the center of human philosophy in that era. The Epicureans were guided by the philosophy that one should enjoy life to its fullest since death is ever near and marks the end of our existence (e.g., I Cor. 15:32). If they could avoid pain and maximize pleasure (both sexual and intellectual), they would do so. They were polytheistic but did not believe any god cared much about the human condition. The Stoics taught the importance of self-denial and recommended indifference to both sorrow and pleasure. They affirmed that the existing universe is god. Both groups were intrigued by Paul's teachings (particularly about Jesus and the resurrection) since they were new to them. The gospel message went against both of these philosophies. It commanded weeping with those who weep and rejoicing with those who rejoice (in conflict with the Stoics), and that we should deny ourselves in reference to ungodliness and worldly lusts (in conflict with the Epicureans).
Although they insult Paul by labeling him a philosophical "babbler" (i.e., a "seed picker"; which literally refers to an idler who makes a living picking up scraps), they provide Paul an opportunity to teach by bringing him to the Areopagus (probably more out of curiosity and an opportunity to ridicule than out of a sincere search for truth). Their motives didn't matter to Paul, however. He would preach the gospel anywhere and to anyone. It is my understanding that the "Areopagus" was both the name of the location (Mars Hill) and the term for the legislative body in Athens that met there. They bring Paul before this group so he can elaborate on the "strange things" he was bringing to their ears. These people were known for this interest in telling or hearing "some new thing." In fact, Luke states that some spent all their time in nothing else but this. What they would hear from Paul would be unlike the things they had heard before. The apostle would not preach human philosophies but divine revelation!