I use to think discussing religion with people that didn't acknowledge the existance of Allah, were hard to talk too. They seemed unreasonable and bluntly ignorant. I was wrong I generalized all of them. Yes, some people are like that. And those people can be extremely difficult to engage in conversation but its because they don't have an ounce of sincerity in them and find satisfaction from argumentation. And than again there are some Atheist who actually use logic and reason. They actually contemplate the world rather than accepting a religion that was handed down to them. In blind belief and not questioning what our real purpose of life is. Islam encourages us to use our logic and reason.
"Zakir, why are you congratulating an atheist?" The reason that I am congratulating an atheist is because he agrees with the first part of the Shahada i.e. the Islamic Creed, ‘La ilaaha’ - meaning ‘there is no God’. So half my job is already done; now the only part left is ‘il lallah’ i.e. ‘BUT ALLAH’ which I shall do Insha Allah. With others (who are not atheists) I have to first remove from their minds the wrong concept of God they may have and then put the correct concept of one true God."
MashAllah. Also before I go on how we should show them the signs of Allah Subhana Wa Talas existence, we really need to re-evaluate our manners. A lot of the times we see Muslims in a dialogue with non-Muslims and they're behavior is absolutely ridiculous. They put the person down and even result in profanity! A major key factor is to talk to them with dignity do not belittle anyone, It will destroy your message and defeat your purpose entirely. Just some advice when you converse with an Atheist you should use the Scientific Miracles of the Quran. Of course the Quran isn't a science book but a guide for humanity. Try to xplain to them how it has not been changed since revelation came down 1400years ago. Unlike many other doctrines that have been changed numerous of times. It contains scientific miracles, just to name a few The big Bang theory, Embryology, the Seas barrier between salt water and fresh etc, etc, etc,. Most of these facts were not even discovered until the 20century.
Everything that exists is an evidence of a Divine Creator, in the Heavens and Earth and even within ourselves. As Humans we aren't perfect but yet this world is, in the since of everything is in a order, in perfect harmony synchrosing as one. The alignment of the Sun and Moon that both follow a distinct course. The transition from day to night. The perfect amount of oxygen and carbon for us to survive. Take for an example your own body, all your organs, nerves and parts are working subject to the laws of nature, and have no choice but to work as they are intended to. How can I present an evidence of God when the signs can be seen everywhere?
"Soon We will show them our signs in the (farthest) regions (of the earth), and in their own souls, until it becomes manifest to them that this is the Truth. Is it not enough that thy Lord doth witness all things?"[Al-Quran 41:53]