In order to effectively judge the rationality of competing viewpoints, it is necessary to first clarify your "Basic Truths"- things you are willing to accept as axiomatically correct. That is to say, things you do not need external justification to believe.
We all have some basic truths we're willing to accept. For example, most people accept that what the can sense through their five senses is real, and not a projection in their mind. They accept that we are not brains floating in vats hooked up to computers, a la The Matrix. Some people do not accept this, and there are a great many philosophical discussions to be had about just this topic (My own opinion is that this is a rather silly debate, as it is in no way positively or negatively verifiable. No presented evidence could alter your view on this, regardless of what your view happens to be, and the question itself is therefore not a particularly meaningful one) Suffice it to say, for the sake of my pursuit of an answer between Atheism and Christianity, I'm prepared to accept this as fact.
It's important also to note that our beliefs often carry with them implicit assumptions that act as axiomatic truths, some of which we don't admit even to ourselves. For example, many of my beliefs as a Christian relied upon the notion that things I have been taught since I was a child deserve the benefit of the doubt. Since I love and respect my parents, there is an implicit trust in whatever they have chosen to pass down to me. On closer examination, however, I came to the conclusion that while my parents themselves are trustworthy, that does not always make them correct. Indeed, things I have been taught since childhood ought to be questioned even more fiercely than the next, because they are the things I have probably accepted with the least critical thought.
Finally, I know from personal experience that these can change for an individual. For instance, I was once prepared to axiomatically accept the existence of God. This is the argument made by many Biblical Apologists (particularly classical ones, like Aquinas and Augustine), and by the Bible itself- that God is plainly evident in the beauty and wonder of the world around us. As we as a human race have gained more understanding of the world, however, what once seemed mystical now seems perfectly rational- moreover, perfectly natural. The processes by which animals, trees, even galaxies live and die are now clear and, for the most part, understood. God, therefore, is no longer necessary to describe and understand the world in which we live. If God is not a necessary part of existence (that is, if we can conceive of a logical framework of existence in which God is absent), then God can no longer be considered axiomatic. My belief is that such a framework exists, and therefore I do not consider the existence of God as a Basic Truth.
So here is my personal list of Basic Truths that I am willing to accept, with the caveat that I reserve the right to change this list at any time. Any future change to this list will be accompanied by an edit describing the change and why it was made
1. I am real
I exist, both as flesh and as a mind capable of conscious, rational though
2. Reason, if properly applied, differentiates truth from non-truth
I say this in a strictly natural sense- I believe that in natural matters, reason and logic can be applied in a meaningful way. That which reason contradicts, cannot be true, and that which reason supports can be true, although is not necessarily true. This, it should be noted, represents an implicit assumption of the veracity of the rules of logic (my definition of reason being nothing more than applied logic)
3. What I sense (observe) is reality
This is covered above, so I won't rehash it here. I will add, however, that what I sense is not necessarily ALL of reality. But it is necessarily at the very least a subset of reality. This also does not imply that I cannot misinterpret or misunderstand my senses, nor does it have any difficulty coexisting with dreams, hallucinations, or mirages. This is simply a claim that my senses are not being faked to my brain in a consistent way (I'm not suffering from some long-term disease that makes me imagine the existence of all other people in my life). Where this gets dicey, of course, is when we get in to the realm of confirmation bias and things that you "feel" to be true. When you "feel" God in you, around you, speaking to you, my view is that this feeling must be treated with the greatest suspicion. I will talk about this in more depth later, but for now let me simply say that this "feeling" is not direct sensory input, but rather a synthesis of something much more complicated.
4.There is a right and a wrong
C.S. Lewis calls this the moral law, or the natural law. It is an acknowledgement that I believe in right and wrong, despite not having an explanation for the existence thereof. This does not connote a perfect understanding of this law, nor is it a claim that all men follow the same law, just that there is some standard that I appeal too, however imperfectly. It is also not clear to me, as Lewis claims, that all men are appealing to the same standard. Or rather, if we are, I have trouble imagining how we could end up with such very different ideas about Right and Wrong. Moreover, it is not clear to me why I consider my own interpretation of this Moral Law superior the interpretation used by others (particularly if we ultimately are appealing to the same standard), but I very clearly do. In regards to the Old Testament itself, I cannot stomach some of the moral actions and decisions- often times morality that is either not condemned or even actively encouraged by the God of the Old Testament
5. Humans are fundamentally different from every other living thing
I would call this the weakest of my axiomatic beliefs. It is the one I find most likely to disappear from this list. It seems to me both fundamentally true and fundamentally absurd that humans are different from every other living thing. The gap between us and the next smartest animal (probably dolphins?) is unimaginably vast. Humans have the spark of something that is unequaled on any scale in the natural world. I suppose this is not so different from claiming that humans have a soul, but that is curiously something I'm not prepared to accept. It's also not so different from the schoolchild, who, upon their first broken heart, is convinced that no one has ever felt what they have felt, no one suffered such poignant loss or cruel fortune as they. It is a self-centered and indulgent conclusion, but one I can nonetheless not, at this point, deny.
6. If Science (Reason) contradicts faith, we are obliged to believe science
I use the word “believe” in the Bayesian sense- we don't actually "believe" anything with certainty, we just accept the theories with the highest probability of being true given our current set of data. This is an important truth, because I think this is where I differ from most Christians, and why so many Christians never end up truly questioning their beliefs. I think if experience, reason, science, and the general weight of human intellect can convincingly show something to be true (or not true), then to continue to hold an opposing belief trespasses into the realm of dogma. Religion and faith are not above challenge. Indeed, if they were, our faith would be childish and arbitrary, for there is no better reason to cling to an irrational Christian faith than there is to cling to an irrational Muslim faith, or Taoist faith, or Buddhist faith. Faith is verified by experience and reason, not the other way around.