Jesus told his disciples to heal the sick, cast out demons and then preach the Gospel. I believe that until or unless we, his disciples do it in this order, we won't be connecting with the atheists or agnostics among us in the post-Christian West. It's the scandal of so many forms of unhealthy living and relationships that keep lots of people from seeing God as creator of the universe and of human life. It's also the awareness of evil in the absence of seeing a divine hand sustaining Being that makes them assume that while a Devil may exist as some malicious alien, God can't.
But we tend to go to the preaching of the story first...then the rules of fellowship....and ignore the call to heal people and cast out evil spirits and wonder why people simply can't understand why we believe in God much less how a man, Jesus, could also be God.
In the Gospels, Jesus goes about healing the sick and casting out demons - so he practiced what he preached. He also empowered disciples to do this "in his name" - and the Catholic saints certainly have been known to heal the sick physically in their lives and after their deaths (via relics, etc.) as well as exorcise or cast out demons either from possession or from oppression, again in Jesus' name.
Do that first and then people will be disposed to listen to the truth claims of the Gospel.
So I go back to the miracles and 'calling cards of God'.... there have been miraculous healings and exorcisms in the recent past and both came about from the prayer of Catholics to Jesus for the afflicted non-believers. There's a reason the martyrs often helped convert their executioners or the witnesses - it was their peaceful, joyful surrender to God's will in dire circumstances that baffled their executioners or enemies and intrigued them.... leading to emotional or psychological healing, the removal of some obsession or oppression, and finally to considering the Gospel truth claims as possible....to finally embracing the faith as their own.
I don't take someone's atheism as an insult. Faith is a theological virtue, a gift, and hence if you don't have it, while you may understand intellectually what we believe, you can't 'own' it or accept it as true until yes, God leaves a calling card in your life.
Sometimes this is as simple as one coincidence too many. Other times it's a genuine miracle that only you would recognize. Other times it's a public spectacle witnessed by others. But it happens enough for us to have confidence that this is how we ought to proceed and should have always been proceeding.