The most obvious answer is that the Gospel is for unbelievers - it is the power of God for salvation. So yes! - let's take the Gospel to all who do not know Christ.
But notice something interesting. Just before Paul tells us that the Gospel is the power of God for salvation (in Romans 1:16), he tells us (in v.15) that he is really eager to preach it. What is surprising is who he wants to preach it to. "I am eager to preach the gospel to you also who are in Rome," he writes. And who is the 'you' he has in mind? He has already told us that at the start of the paragraph when he thanks God for "all of you, because your faith is proclaimed in all the world" (v.8). He is very keen to preach it to 'you' believers in Rome - the one whose faith has been talked about all over the place, the ones who he says are "loved by God and called to be saints" (v.7).
This leads us to a clear conclusion. The Gospel is supposed to be preached to believers. Paul is still not sure when he is actually going to be able to follow through on his plans to travel to Rome, so his bursting desire to preach the gospel to the church there drives him to take out a pen and start preaching it on paper right away. For sixteen glorious chapters he proceeds to preach the gospel to believers.
But some would object: "We already know the Gospel! We heard it and believed it years ago. Tell us something new. Give us some practical instructions on how to live the Christian life." Certainly Paul will eventually demonstrate that the preaching of the Gospel has practical implications. But he is clear: Christians don't need another "How to..." book to add to the overflowing shelves of such books in the Christian book-store. They need the Gospel.
If we are not interested in hearing the Gospel for the three hundred and twenty third time, we have a problem. Paul tells us that he is writing as "a servant of Christ Jesus" (v.1). He is not writing on his own behalf or sharing his own insights and opinions. If we are not interested in what Paul has to say, we are not interested in what God has to say - for Paul is only a servant. And Paul was set apart for one reason alone - "for the Gospel of God" (v.1).
Paul did not have a repertoire of powerful sermons relevant for every occasion. He did not have a catalog of seminars to enable believers become successful in various aspects of their Christian life. He just had the Gospel. No list of tips on how to be victorious or holy or popular or healthy or powerful. Nothing much about what we have to do. But a wonderful Gospel about what God has done in Christ. If Paul reflected God's heart (as he surely did), God is eager for the Gospel to be preached to the church. After all - what else is there to preach?