Where to start in your discussion is important. Muslims do not see any flaws in their faith, so you have to understand where they could be most receptive and/or most impressed. The first filter, receptivity, is important because Muslims run from challenging discussion faster than any other group I've seen. Most Muslims would rather live with a lie than confront any illogic in their faith. The culture is very different than other faiths and the root is in the Quran:
YUSUFALI: Already has He sent you Word in the Book, that when ye hear the signs of Allah held in defiance and ridicule, ye are not to sit with them unless they turn to a different theme: if ye did, ye would be like them. For Allah will collect the hypocrites and those who defy faith - all in Hell:-
PICKTHAL: He hath already revealed unto you in the Scripture that, when ye hear the revelations of Allah rejected and derided, (ye) sit not with them (who disbelieve and mock) until they engage in some other conversation. Lo! in that case (if ye stayed) ye would be like unto them. Lo! Allah will gather hypocrites and disbelievers, all together, into hell;
SHAKIR: And indeed He has revealed to you in the Book that when you hear Allah's communications disbelieved in and mocked at do not sit with them until they enter into some other discourse; surely then you would be like them; surely Allah will gather together the hypocrites and the unbelievers all in hell.
Above you can read three translations of 4:140 - they all say the same thing. Don't talk to those who reject or deride the Quran. It is in the culture to run from conversations that challenge the Islam. Why? Well, the Quran answers that too with circular reasoning:
YUSUFALI: Do they not consider the Qur'an (with care)? Had it been from other Than Allah, they would surely have found therein Much discrepancy.
PICKTHAL: Will they not then ponder on the Qur'an? If it had been from other than Allah they would have found therein much incongruity.
SHAKIR: Do they not then meditate on the Quran? And if it were from any other than Allah, they would have found in it many a discrepancy.
You see there are no discrepancies because the Quran says so. So not only must you not speak with those that challenge the Quran, but you shouldn't even ask the questions yourselves:
YUSUFALI: O ye who believe! Ask not questions about things which, if made plain to you, may cause you trouble. But if ye ask about things when the Qur'an is being revealed, they will be made plain to you, Allah will forgive those: for Allah is Oft-forgiving, Most Forbearing.
PICKTHAL: O ye who believe! Ask not of things which, if they were made unto you, would trouble you; but if ye ask of them when the Qur'an is being revealed, they will be made known unto you. Allah pardoneth this, for Allah is Forgiving, Clement.
SHAKIR: O you who believe! do not put questions about things which if declared to you may trouble you, and if you question about them when the Quran is being revealed, they shall be declared to you; Allah pardons this, and Allah is Forgiving, Forbearing.
YUSUFALI: Some people before you did ask such questions, and on that account lost their faith.
PICKTHAL: A folk before you asked (for such disclosures) and then disbelieved therein.
SHAKIR: A people before you indeed asked such questions, and then became disbelievers on account of them.
Quran barely read and Arabic myopia
Now, most Muslims have not read much of the Quran in their native language even if they speak Arabic so they won't run from conversation because of the above quotes. They will run because the culture has already been created by those quotes. Funny, in fact, that if you don't want them to run - be as direct in your rejection as possible.
Many of those that have read the Quran, are non-native speakers following this doctrine from Mohammad:
"Verily the one who recites the Qur'an beautifully, smoothly, and precisely, he will be in the company of the noble and obedient angels. And as for the one who recites with difficulty, stammering or stumbling through its verses (because he doesn't know how to read it but is trying to), then he will have TWICE that reward."
However, even if they memorize the Quran, they usually do it in Arabic. On a side note, it is important to remember why - that the Quran places Arabic above all languages:
YUSUFALI: We know indeed that they say, "It is a man that teaches him." The tongue of him they wickedly point to is notably foreign, while this is Arabic, pure and clear.
PICKTHAL: And We know well that they say: Only a man teacheth him. The speech of him at whom they falsely hint is outlandish, and this is clear Arabic speech.
SHAKIR: And certainly We know that they say: Only a mortal teaches him. The tongue of him whom they reproach is barbarous, and this is clear Arabic tongue.
The Quran goes on further with a little racism:
YUSUFALI: Had We revealed it to any of the non-Arabs,
PICKTHAL: And if We had revealed it unto one of any other nation than the Arabs,
SHAKIR: And if we had revealed it to any of the foreigners
YUSUFALI: And had he recited it to them, they would not have believed in it.
PICKTHAL: And he had read it unto them, they would not have believed in it.
SHAKIR: So that he should have recited it to them, they would not have believed
So...back to the point...where to start
When a Muslim starts telling me about the "scientific miracles" in the Quran, I challenge them to send me a single one. Simultaneously, I ask them to explain a few of the top mathematical, historical and scientific errors.
With female Muslims, especially those with kids, I usually ask them if they're ok with the sometimes violent sexism or with Mohammad's pedophilia. This tactic sometimes works with men, but you'll be surprised at how many men are ok with these issues. More on that later.
There are many other passages of the Quran or Hadith that would have most of us either laughing or in horror, but I promise you that many of them will have little effect on the average Muslim. These include passages of torture, murder, unscientific nonsense, racism, Antisemitism, etc.
In the next post, I'll start with key quotes from the Hadith on Mohammad's pedophilia. Although this is not in the Quran (except for some strange verses about young boys and wide-eyed girls in heaven), it helps to go here quickly because it often comes up right away. Again, though, I don't recommend this issue as a starting point for all discussions.