There are two distinct intellectual battlefronts when debating Judaism, the Ultra-Orthodox position and the Modern Orthodox position. (Obviously, both these battlefronts exist on a spectrum. There are some people in the Ultra-Orthodox camp who hold pretty modern positions regarding important issues. Additionally, there are some in the MO camp who hold UO positions on a number of issues.) To provide some clarity, here is the basic breakdown:
The Ultra-Orthodox Position:
Evolution: Science is wrong. Adam and Eve were the first humans and they lived around 300 generations ago. Evolution is debunked stupidity. We do not come from monkeys.
Ago of the universe: The world is 5,781 years old. Science cannot talk intelligently about the past. It is all just a theory that people made after looking at some old rocks.
The flood: The Mabul covered the whole world. It was all a big nais, so there are no questions.
Chazal: Chazal had divine knowledge. Chazal certainly did not make mistakes about science - especially when it involves knowledge with a halachic contingency. Magic and demons are/were real. The medical cures don't work because the secrets of how to administer them were hidden away. Most midrashim are literal unless there is a really good reason to think otherwise. The fact that a midrash contradicts science is not a good enough reason. Period.
Main proofs for Judaism: (1)The Kuzari argument; (2) A more vague "our unbroken mesorah"; (3) fulfilled prophecies; (4) stories of gedolim performing miracles; (5) and statements of Chazal that show divine knowledge.
Moral issues: Moral issues are irrelevant. God gave the Torah and it is the source of all morality. Not some American liberal narrishkeit.
Biblical scholarship: Not really relevant. The Torah was written word for word by the directive of Hashem. He can write in whatever style he wants. You cannot ask questions about how He writes.
Archology: It is all speculative silliness and they do not know what they are talking about. Also kings back then did not write about their losses.
Main Players: Rabbi Dovid Gottlieb, Rabbi Avigdor Miller, Rabbi Moshe Meiselman, Rabbi Lawrence Kelemen, (Rabbi Alexander Hool, Rabbi Dovid Sapirman, Rabbi Reuven Schmeltzer)
Modern Orthodox Position:
Evolution: Science is correct. Evolution happened. Adam and Eve is either a parable or the first people imbued with the divine spirit.
Age of the universe: The world is billions of years old. The first chapters of Bereishis are not literal.
The flood: The flood was local. or, alternatively, the flood never happened and the Torah is recasting the story of Gilgamesh with its own twists.
Chazal: Chazal did not have divine knowledge of science and their statements reflect and sincere attempt to understand the world given the science of the time. Nevertheless, halacha, in most cases, should not change. This is because of the way Hashem created the Halachic process. Demons are bacteria, mental illness, etc. (or, alternatively, Chazal were mistaken about their existence). Magic is not real. Midrashim are almost entirely non-literal. This is true even if there is a halachic contingency dependent on the literalness of the Midrash.
Main proofs: (This is where things start going sideways for the Modern Orthodox position)
Step 1: Fully embrace the post-modernist idea that "knowledge" and "proofs" are silly constructs.
Step 2: Discuss how the Jewish people have a unique history and impact in the world. Combine that with talk of a Sensus Divinitatis ("sense of divinity") as well as mix in some form of "mesorah" argument."
(Additionally, The Kuzari is a weak argument that would never be accepted in the academic world; The statements of Chazal that show divine knowledge are wrong)
Moral issues: Moral issues are massively important and the moral problems in the Torah are must be mitigated by claiming that (1) The laws were only for back then but we are expected to outgrow them and embrace true morality, (2) emphasize the moral framework undergirding various laws (darchi noam, ayva, etc.).
Biblical Criticism: Hugely relevant. Issues are dealt with by pointing to the lack of consensus amongst scholars to undermine the field as well as point out the speculative nature of the stories. Additionally, this issue is dealt with by acknowledging that parts of the Torah may not have been written by Moshe, but that a careful examination of the 13 principles of faith show that such belief is not required.
Archology: Also a huge issue. Generally dealt with by (1) insisting that a lack of evidence is not evidence, (2) allegorizing any falsified date or population number in the Torah, (3) Re-interpreting the pesukim to accommodate the archeological findings.
The main players:
Rabbi Natan Slifkin, Rabbi Joshua Berman, Rabbi Amnon Bazak, Rabbi Jonathan Sacks, Rabbi Lopes Cardozo
Final Thoughts
Both these positions have strengths and weaknesses. The UO world retains adherents by enforcing ignorance and caricaturizing outside knowledge. As long as there is strict policing over internet access, university attendance, and interactions with non-Jews, most UO people spend their life blissfully ignorant of anything that will shake their faith. Large walls are built around the community, and by frowning on birth control, the UO world simply out-breeds any potential attrition of disenfranchised members. That said, for someone questioning, to "debunk" the UO position, all one really needs to do is realize that it is eminently more sensible to believe the findings of the academic world over the opinions of a few outliers. From there everything crumbles. By the time one starts having "sefakos" it is almost always already too late.
Conversely, the MO position presents a far more difficult target. They do not hide from the world and are proud to be considered members of the lager global and academic communities. The questioner is told to accept all the science they want, everything can still be interpreted in a way that allows for belief.
However, from my perspective, the MO approach, while great at defense, sucks at offense. On this forum and in books written by MO people, I find that they cannot adequately explain the simple question: Why do you believe? What is your actual positive evidence to accept that the bible is anything other than a work of fan-fiction? I am curious about how people here respond to that prompt. Do MO people have any strong reasons to believe that a super-being, the creator of the universe, in fact, came down our parents a book full of rules about bloodstained underwear, animal sacrifices, and stories about giants and talking snakes?
Religion is a powerful force, but I do not think that either the MO or the UO position is sustainable in the long term. The walls are crumbling around the UO world and exposure t the outside world is kryptonite to their intellectually indefensible positions. The MO position is equally precarious. They are raising thier children in a libiral world that simply does not tolerate the views espoused in the Torah. Excousies and creative reinterpretations can only go so far. Without any compelling reason to remain a "member of the tribe," the next generations will simply fade away into the larger world.