No matter where he went, Paul sought out people to share the good news of Jesus with. He did not limit himself to the synagogues, but he also went to the marketplaces to speak to those that happened to be there. Even in Athens, an epicenter of learning and discussion, he went to the places where people met at leisure. He initiated discussions; he was not ranting on a soapbox or lecturing others.
This is a great model for spreading the good news in this day and age as well. It does not have to be a confrontation, it is actually better if it is not one. Striking up conversations and gently leading to discussions about God and the path of Jesus lends itself to not shutting the other person down. We should be willing to listen more than we talk, too. In my experience, if you pay attention and show interest and compassion, others will start asking your opinions and what you believe, and why, without having to be aggressive in your approach.
I know that prior to becoming a Christian, anyone approaching me with the ‘patented evangelical aggressive approach’ would cause me to look for the nearest escape route. Even now I occasionally get subjected to this and now I find I am embarrassed at the hammering style of beating Jesus into the hearts and minds of others. It rarely works, and even when it seems to be working, the victim is usually just saying whatever they have to in order to be left alone until they can di di mau. Been there, done that.
Sharing our own stories when asked shows that we are not holding ourselves above, but rather admitting where we have done wrong, and allows others to identify with us. It was hard for me B.C. (before Christ) to find common ground with someone presenting themselves as a saint from the word go. Once that connection has been made, people perk up when they hear how your life has been turned around and made new again. Be honest and unafraid to admit if you do not know the answer to a question. And if you face rejection by the person you are talking to, do not take it personally or get offended. As Christians, we should be the hardest people to offend. Just thank them for their time and either change the subject if the opportunity to continue the conversation presents, or move on.