Basically Paul goes to Thessalonica and spends several seeks among the Jews there, talking in their synagogues to try and persuade people that Jesus is their awaited Messiah. In one of my favorite translations in the NIV, some Jews were opposed to Paul and "rounded up some bad characters" to riot against Paul, basically running him out of town. Paul goes to Berea and repeats his message, where more people listen, but the same troublemakers come and force him out again. Finally Paul goes to Athens and, while waiting to be joined by his brothers in Christ Timothy and Silas, starts conversing with the Greek philosophers, who invite him to explain his teaching at the Aeropagus (also known as Mars Hill, the namesake for our discussion group). He then presents a quick, eloquent explanation of the gospel, using the Greeks' devotion to their religion and their altar "to an unknown god" as a starting point.
Several points can be taken from this passage. Paul's commitment to his mission (and his message) is made pretty clear after he keeps preaching the gospel after being repeatedly run out of town and threatened by some "bad characters". In 2 Corinthians 11, he goes into more detail about the troubles he's been through for the gospel: imprisonment, flogging, being shipwrecked, going hungry, and the concerns that come from being responsible for churches all around the Mediterranean. This Christianity thing is clearly much more than a social club where people share feel-good stories and sing songs. No one dies for a social club.
Besides all the threats, Paul also perseveres amidst discouragement. Each time his message has some believers and (often a lot of) detractors. He doesn't convince everyone. This mirrors evangelism today where it's so easy to think we're not accomplishing anything, that no one is listening. It's a matter of faith to believe that God really is making a difference through our proclamation of His message. Paul also doesn't seem to have much time to start any kind of formal, established church in any of the three places. Of course, a church in those days was basically a big Bible study meeting in someone's house to worship God, a bit easier to set up than a modern church. The important thing for Paul was simply to help people understand God and the gospel, and train leaders to steward the church after he left.
One other thing we can get out of this chapter is how Paul makes an effort to speak to his audience out of their culture, in a way that makes sense to them. In the synagogues he speaks from the Jewish scripture to prove that Jesus is the messiah. On Mars Hill he builds off of the Greeks' religious practices and even cites one of their poets. He's attempting to make his message less foreign and more credible by linking it to sources and traditions his audience considers reliable. This kind of cultural awareness and willingness to adapt the delivery of the gospel to your audience is no less important today.
But though Paul adapts the way he preaches the gospel to make it more palatable to his audience, he never adapts the message itself. He doesn't try to hide or obscure the parts of it that will be difficult of might turn people away; for the Greeks it was the resurrection, in modern culture it tends to be judgment, hell, or even Christianity's claim to absolute truth and the only way to heaven. You might win more people over to a diluted, feel-good version of Christianity, but is it a victory if they don't know the real God? Going back to my emergent church post, it's smart of them to be aware of postmodernism and try specifically to speak to this culture, but diluting the gospel with the fuzzy truth and relativism of postmodernism is dangerous.