I’m not keen on the word ‘prove’. To me, proof is a mathematical or logical demonstration which achieves an incontestable conclusion from some uncontested premises. The trouble is that in normal conversation premises are very rarely uncontested, and the arguments drawn from them rarely incontestable.
However, the OP itself, other than the title, does not ask for proof. It asks for “sound rational arguments”, which is much more approachable. Still, if we are to avoid the pitfalls often mentioned above - arguments which go along the lines of ‘Christianity must be true because the Christian scriptures say it’s true’ - then we need to be very clear about our definitions, and what, exactly, we want demonstrated. Does the OP demand that Christianity is exclusively true, or that it is among many truths? When Rose asks: “Does it really matter which religion is followed” she implies that she is looking for arguments for exclusivity - that Christianity is true and Buddhism, say, or Shinto, is false. This has been addressed, albeit somewhat half-heartedly, by numerous popes. Salvation, say any number of encyclicals, is attainable by people who are outside of the Church, if they are excused through ignorance beyond [their] control. This is the concept of ‘invincible ignorance’, very well explained by Petra above.
So perhaps we are attempting to demonstrate that Christianity is at least one valid way of achieving eternal life. I think that’s probably easier than demonstrating that it is the only way. But now we must decide what we mean by Christianity. Is Roman Catholicism more valid than Orthodox Christianity? Or Methodism, or Mormonism?
It is interesting that C.S. Lewis, whose works are recommended above, specifically rejected Catholicism. “To us the terrible thing about Rome is the recklessness (as we hold) with which she has added to the depositum fidei – the tropical fertility, the proliferation, of credenda.”
Still, on a ‘Catholic Answers’ forum I suppose we must be thinking of the Roman Church, especially as much reference has been made here to the depositum fidei so disparaged by ‘Protestants’.
Christ was fairly concise about the essence of his message: Love God and love your neighbour. “On these two commandments hang all the law and the prophets.” The OP rather echoes this: “does it really matter which religion is followed as long as people try their best to be good people?”, so it sounds as if she has already “accepted Christianity…”. So what does she mean by going on with “… in all its fullness”?
I’m guessing she’s asking whether we can demonstrate that Christ was the son of God, or that he rose from the dead, or that the depositum fidei are valid. As it happens, I think we can, although we will have to start with a clear understanding of what we mean by “the son of God.”