First, you should know that I was an atheist for about 10 years, so I definitely feel where you’re coming from and I have some idea of what you’re going through.

You’ve brought up a few different things. This is part of what I meant by moving goalposts. A more systematic approach would be better, I contend. I suggest breaking this down: 1.) proof of the supernatural and 2.) proof for the existence of God and 3.) proof that the Catholic worldview is the correct one

1.) Miracles would at least prove supernatural. Depending on the miracle, it could be on piece of evidence for the existence of not just supernatural, but of God. Alone it would not be sufficient to prove but add up many miracles plus logical proofs and I’d say you have more than reasonable evidence.

The Catholic Church only acknowledges miracles like you describe — no other natural or scientific explanation after thorough examination by multiple experts. More reading if you’re interested:
https://en.wikipedia.org/wiki/Canonization#Catholic_Church
http://www.economist.com/node/304212

And these are just miraculous medical miracles by saints, I haven’t included the eucharistic miracles or the incorruptibles.

2.) Proof for the existence of God — I could go into a lot of these and some of them are quite serious and some of the supposed refutations by atheist websites are rather lame or miss the point of the proof entirely which is a shame. But one of the best ones I’ve heard is: Why is there something and not nothing? And note, I mean nothing — total nonexistence. Not Dawkins’ “space dust” or whatever he said preceded the Big Bang, I mean lack of anything. Why does the universe exist pre- or post-Big Bang?

Sufficient study into that question will lead you, inexorably, to the conclusion that there must be an uncreated creator, an uncaused cause, or Aristotle’s unmoved mover.

3.) Proof for Catholicism’s claims — this obviously depends on (1) and (2), so it might be premature to discuss this part as it’s more involved. You have to walk through history and see how God’s revelation points to Christ and how Christ established the Church. There is overwhelming historical evidence for the Catholic Church being the original Church that Christ and the Apostles founded. I can go through it all but, like I said, it may be premature