Since most atheists require evidence, what would be the best way to approach them, aside from using scripture?
Why not the way Paul approached the Athenians (Acts 17), or the Lystrans (Acts 14)? There he speaks from what they are already familiar with—nature and culture.
In our day we may also want to reason from cosmology (the surprising scientific discovery, last century, that the universe has a beginning—the world didn't exist "before" a time thought to be 13.8 billion years ago.) We could reason from the design of the universe—the values of at least 50 parameters seem perfectly "fine-tuned" to lead to intelligent life. We could also reason from morality. If there is no God—no higher power somehow outside our world—then nothing is truly, absolutely right or wrong. Morals and ethics are only relative.
All these lines of thought are readily available at the internet. Or maybe check out some of the material at this website. (Or in some of my public debates.)
Of course what will eventually capture hearts is the story of the Bible and the person of Jesus. All of us can locate ourselves somewhere in the grand narrative (creation-fall-redemption-new creation). And when we see Jesus Christ close-up, we cannot help but be moved, if we are receptive—truly seeking (John 12:32; Matt 6:33; 7:7-8).