I have been recently studying how to be a more effective witness. I purchased a course from Don Johnson ministries that I highly recommend. I am still just learning the material. The first thing to do of course is to pray. Then it is usually better to listen first and ask them questions. Rather than to try and force a conversation down their throat. Then you can enter into a dialogue based on what they say while still directing the conversation from being side tracked. You could for instance ask them if they are truly happy. If they are honest they will usually say they are not.
A religion (and even atheism) is really a world view. So you are really dialoguing with people who have different world views. The goal should be truth. Since it doesn’t matter how much someone may like their world view if it is not true then it should be abandoned for the truth. Which world view better explains the data or corresponds to the evidence. Don’t get bogged down in the details or objections to the faith. You want to look at the big picture. Does your world view hold up to reality? People who change religions or their world view often do so because they realized their world view did not hold up to reality. Jennifer Fulwiler a convert from atheism tells how her atheist world view started to break down after the birth of her first child. She could no longer see how atheism could account for the love that she was experiencing for her child and in this new family she was a part of. She just knew that there was more to love than just chemicals in the brain. She started to realize there was more to life than just matter. As she dug into the Catholic world view she realized it made the most sense of reality.
At any rate asking if someone is happy can lead into a conversation that is directed to showing them that God is what is missing in their life. Personal testimony about how God has changed your life and others is important here and is one piece of evidence. Jesus is the living water that will satisfy our souls. And until a person has Jesus it doesn’t matter how much sex, cars, material things, relationships, etc they have they will never be satisfied because the world is not enough. We were made for more than this world. We were made for God. As Peter Kreeft says never trust a happy atheist. This is where we appeal to their sense that something is missing in their life that only God can fill. If all of your natural desires can be met like your desire for food or your desire for sex then what about the desire that the world can not satisfy? Would it make sense to have a natural desire for something beyond the world if that desire could never be fulfilled?
Miracles and supernatural experiences. These can be another line of evidence. Skeptics who reject miracles outright do not do so on the basis of evidence but on the basis of a presupposition that there are no supernatural explanations. It is actually difficult to maintain such a view because one has to continually deny reality in the face of ever new mounting evidence to the contrary. Constantly new miracles happening all the time for instance. Do they look at all tbese miracles and the evidence? No, they are rejected not based on evidence, but on presupposition. Famous atheists will even admit that it seems like the universe is designed and it seems like God is the explanation, but in order to justify their desired world view they must be stubborn to their presupposition.
Most of us have some supernatural experiences in our lives if we look back and reflect. Sometimes people forget them. If you ask them they may remember this one time when they had this experience. It could be a way to get them to reflect on God working in their life. As well as share some of your own.
The reason many skeptics reject Christianity is not because it doesn’t make sense but because they have sin in their lives. And, they do not want to change. And, a lot a times especially in our day and age it is sexual sin. Scripture says that God gave them over to perversion where they exchanged the glory of God for the lust of the flesh, for unnatural relations. Sexual sin is a big sin because it denies our purpose and drives a road block between us and God. These are the deeper reasons behind the objections. People do not want to accept reality as it is. They want to create their own reality. Instead of doing things according to how God created things to be, they want to do it their own way. Sexual sin represents a deeper rejection of God. Sex is meant to be a reflection of God’s love, to give ourselves as a self Gift to the other. Sexual sin is a denial of that and distorts that truth. It turns sex into an act of self gratification. It denies the purpose of who we were made to be by God.
It is usually for emotional reasons they reject God and then afterwards try to justify their decision with their reason. But, in order to do so they must deny reality as it is and make it fit with their new world view.
In my own experience if the Holy Spirit is leading the way and the person is prepared to hear what you say it is relatively easy. If not then it is an uphill battle. I had this one time where I was working on a job and someone asked me about Jesus. I don’t even remember much about what I said. But he was ready and he accepted Jesus right there. But I had this other experience where I gave this stranger a ride to his house. He was really friendly and everything but as soon as i mentioned God he completely changed. it didn’t matter what I said he would have none of it. Ultimately, it is the Holy Spirit that makes it happen. But if the person’s heart is hard to God then all you can do is pray for them. Listening to them first may give you an opportunity to speak after. But it may take years to make any head way if at all.