Hello. This is first topic that I'm actually get into discussion so I hope I don't mess things up. Before I say anything I just want to note that I'm 17 year old teenager so maybe I won't be able to point to a lot of things that matter, but I will try my best to do it. Now, I'll post the way I had "expirience" with, but if you want to get deeper into that I suggest you start researching apologetics ("religious discipline of defending or proving the truth of religious doctrines through systematic argumentation and discourse." - source Wikipedia). Lot of video resources can be found on this youtube channel
Theology, Philosophy and Science
Anyway this is how would I do it. I think that first we as Christians should do is to just reaveal the Gospel to them. That by itself won't be enough to average atheist since they probably think: "You actually believe in that?". But never mind that. The point is that if they are willing to listen to you long enough and open mindedly observe all the evidence that you can provide, they can change. I want to repeat what others have already said here and that is that it can only be done with the blessing of the Holy Spirit but I'm sure that God will be with you while you speak
OK, so let's get back to the topic.
Now, I think that is easier to introduce God to people is through Jesus than with Old Testament and all that cosmological evidence stuff. I mean, it can be done that way as well but I find this way better because Jesus was physically present right here on Earth as a man.
Now to talk this to them I would go this way:
1. There is historical evidence for existence of Jesus (you can research this more but I will just say that Tacitus which is one of the most important Roman historians is showing evidence for Jesus on one page is his Annals).
2. Now, since we know that Jesus is real and that he existed we need to make them find out that things he did was real as well. That can be done by showing them
evidence that the New Testament is reliable. Now, I believe there are lot of people who researched this, and this is one of the reasons lot of atheits converted to Christianity. One I used to find this out is J. Warner Wallace which was atheist for about 30 years and after researching the reliability of the New Testament changed to Christianity. You can find his lecture on youtube channel that I linked above or you can read his book named Cold Case Christianity.
3. Now after you showed them evidence on Jesus being real and New Testament being reliable which is prooving that things Jesus did was real as well, you can start talking about the Old Testament since Jesus said: "Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them." Matthew 5:17. This means that if New Testament is True than that automatically means that Old Testament is true as well.
4. With all the evidence showed until now, they should be able to find out the truth and change to Christianity. If this still is not enough, there are more evidences to show. You can show them some of this stuff:
* Chariots in Red Sea Chariots in Red Sea: ‘Irrefutable evidence’
* Actuall sulphur balls from Sodom * Some of the popular atheists changing, one being Antony Flew (do a bit on research here if you want to but I'll just mention name of his book here - There is a God: How the World's Most Notorious Atheist Changed His Mind)
* Darwinian evolution theory is false: You can find a lot of material on youtube channel I linked on top of this post.
*And of course, TESTIMONIES: Now, this is the ultimate evidence. Testimonies from people who expirienced God through miracles. This is real stuff. You can find dosens of material here but, again, here is another youtube channel with some of the stuff you can show them
Living For Christ
There you go my friend. I hope I managed to cover lot of important aspects on this subject. Of course, I know there are much more to show but I think that if anyone really want to convert this should be more than enough for them. It's a bit longer post but I think that you should be able to find most of the stuff you need on sources I provided, and if you don't, Google is preety big
Anyway, here you go. I hope God helps you on your journey and remember
we can do all things through Christ who strengthens us!