(No affiliation. They make great stuff. All the links are their property.)
Living Waters is a ministry that both does and teaches evangelism. They do it public places, one-on-one, and mostly with strangers. They've helpfully put up many videos for you to learn from. Ray Comfort is the main evangelist in their videos. Typing his name into Youtube with words like evangelism will show real conversations.
The video Hell's Best Kept Secret dives into their beliefs about evangelism. I recommend watching it. Another is Evangelism is as Exciting as a Root Canal.
There's many theories on how to do evangelism. Quite a few have you work long term trying to build up opportunities to share, drop hints, or whatever. First, we don't see Jesus and the Apostles doing that. We're to learn from them. Second, the people you're delaying that conversation with might die before you have it. Only God knows how much time they have, He told us to share His message, and attempting it quickly makes sense. How do Jesus and the Apostles do it?
They pray first to rely on God. They open a conversation with someone. At some point, they use the Law to convict their hearts about their sin. They make it personal by letting their audience know they, not just people in general, are guilty of these things. That creates godly sorrow that's necessary for real repentance. They're told of the eternal consequences. Once they know their problem, the evangelist shares the Good News where Jesus Christ solves the problem they have.
God's Word predicts the Gospel will have a strong effect on people when we're doing it right. Most will be offended by Jesus. Whereas, the Holy Spirit will convict those He's pulling toward Him. My favorite part of Ray's videos is that you can see this in their eyes. If not eyes, tone and body language. These are good things to watch for if you're out sharing the Gospel. Have faith that God is always at work but seeing it is encouraging.
What Biblical Evangelism Really Looks Like
How to Witness to False Christians
Jehovah's Witness Clashes with Christian Evangelist
Their training and/or videos are a good investment. I suggest memorizing their lines for getting to the Law, delivering it, the Good News, and helpful metaphors. God might also inspire you to make your own that suit your style better.
Living Waters' site: Basic Training. Some conversational tactics to memorize. Their book was good. Maybe use their training in discipleship groups.
They have a School of Biblical Evangelism. If a member, lesson PDF's are here.