Much could be said about manuscript evidence (demonstrating conclusively that the Bible has been reliably passed down to us), archaeological evidence (demonstrating that the locations, lifestyles, and many of the people in the Bible are genuine), eyewitness testimonies and changed lives (demonstrating the genuine sincerity of the authors, such as the apostles Peter and Paul, who were executed for their claims), and fulfilled prophecies (such as Daniel 9:24-25, Leviticus 26:44, Jeremiah 32:36-37, Ezekiel 34:12-13, Micah 5:1-2, Isaiah 53, Zechariah 12:10, Ezekiel 26:2-5, Luke 21:7-24 demonstrating the supernatural inspiration of the authors).
However, the most basic proof of Christianity, and thus the reliability of the Bible that defines it, is that, to rephrase the words of the late Greg Bahnsen, without the God who is most clearly revealed in the Bible, it would be impossible to prove anything at all.
Atheism is logically ruled out because under Atheism, one must conclude that past time was infinite in duration, or that nothingness created everything.
If we choose to believe that past time was infinite in duration, we are claiming that, since past time has ended, we have reached the end of infinity. Yet infinity, by definition, has no end. Thus, this view is logically impossible.
If we choose to believe that nothingness created everything, then this nothingness acted volitionally and voluntarily out of its own self-direction and self-motivation (since there was nothing around to force the nothingness to create), which makes it a "personal" nothingness. Since the nothingness initiated time, it must inherently transcend time, making it eternal. Since it has caused absolutely all things that have ever been, the nothingness is also necessarily omnipotent. Apart from Atheists, when people think of eternal and omnipotent persons, we don't refer to them as "nothingness", but rather as "God". Thus, this view, that nothingness is equivalent to somethingness, is also logically absurd.
The only logical possibility is that a personal, eternal, omnipotent God initiated time.
This not only rules out Atheism, but also any worldview that proclaims that the ultimate authority is NOT omnipotent, NOT eternal, or NOT personal; or that past time was infinite in duration. These worldviews, such as ancient Nordic views, ancient Roman views, ancient Grecian views, Wicca, Buddhism and Hinduism, are logically impossible.
Since this omnipotent, eternal, creator God caused us to have consciences and to recognize the existence of objective morality and objective justice, and since there would be no such thing as objective morality or objective justice if this omnipotent creator God were amoral or unjust, it is logically necessary that this eternal, omnipotent creator God is also both inherently moral and inherently just.
However, if God were uninvolved with our world and uninterested in our affairs -- if God simply didn't care, then there would be no such thing as objective justice and nothing could be considered objectively just or objectively unjust, for there would be no ultimate authority to judge it as such. Justice would never be served and God would then be unjust, negating the possibility of objective justice. Yet we know that certain things are in fact inherently unjust. This establishes the impossibility of any worldview or religion that denies an immanent God, such as Deism or Zoroastrianism.
Because so many religious views are logically impossible, any worldview that teaches the unity of all religions must also be logically impossible. This would include the Bahai Faith, the Unity Church, and any other form of universalism.
We are left with only the worldviews that claim an omnipotent, eternal, unchanging, immanent, personal God, who is inherently moral and inherently just. Everything else is logically impossible.
Any worldview that denies the sacrificial atonement made by God Himself also denies the possibility of objective justice unless it damns all sinners and forgives none, for to forgive sins without atonement is simply to ignore justice. This renders Islam logically incoherent, as the God of Islam claims to be inherently just (Surah 4:40, 2:272, 2:281, 3:18, 3:108, etc), yet when he shows mercy to sinners (Surah 4:64, 4:96, 4:100, 4:106, etc) He ignores His own justice. So He is just and not just at the same time and in the same way.
The God of the Bible never does this. Perfect justice, according to the Bible, was wrought upon Jesus, the Son of God, who took our punishment in our place (1 Peter 2:24). Our fines do not go unpaid in Biblical Christianity, as they do in Islam. In Christianity, Jesus paid our fines for us on the cross. In Islam, our fines remain unpaid, and justice is never to be served.
Modern Judaism can also be ruled logically impossible since their own scriptures, such as Daniel 9, Isaiah 53, Psalm 22, and many others, prophecy of Jesus of Nazareth as the coming messiah, with Daniel 9 even stating quite clearly and accurately when he would come, and yet modern Judaism rejects Him, thus rejecting its own scriptures.
If any of the above worldviews (Atheism, Wicca, Hinduism, Deism, Zoroastrianism, Bahai, universalism, Islam, Judaism etc) were true, then the laws of logic, specifically the law of non-contradiction, would have to be rejected. The law of non-contradiction states that nothing can both "be" and "not be" at the same time and in the same way. All of these worldviews claim something in their most basic tenets that is logically contradictory, as we have already seen. Thus, for any of them to be true, it would be necessary to conclude that the logical law of non-contradiction is not valid. This would ultimately mean that things can both be and not be at the same time and in the same way, and that knowledge itself is impossible, and that therefore, as Bahnsen said, it would be impossible to prove anything at all.
So we see that Christianity is necessarily true simply because of the impossibility of the contrary.
Personally, I am convinced of the truth of the Bible simply because I have met Jesus, the same Jesus revealed in the Bible (John 20:29). His Holy Spirit has brought His words to my remembrance at the right times and in the right circumstances (John 14:26).
His word has been confirmed to me, and, since we have seen already that an omnipotent, eternal, immanent and personal God must exist because of the impossibility of the contrary, you have every reason to believe that His word can be confirmed to you as well.
Additionally, Hinduism just as perfectly fits your description of the kinds of religions which are the only ones logically possible. Its personal, immanent, omnipotent, moral god is Brahman. He is the eternal spirit which initiated time. The universe is as a dream to him. In addition, he did so in such a way that reincarnation would naturally mete out justice upon wrongdoers until they are pure of heart and become one again with Brahman.
I don't mean to suggest Hinduism is better than Christianity, or that it should be considered at all for that matter. I personally reject each based on different flaws within their worldviews.
On a side note, because you originally listed Hinduism as a religion without a personal, immanent, eternal God, I assume you're unfamiliar with it to some degree. I recommend you verify that what I've said about Brahman here is true, and also think a little about how familiar you are with a religion before you start bandying about claims about whether it meets certain criteria. Arguing against a viewpoint is much easier when you have accurate information about it, so reading up on other religions is just as important as reading up on your own.