Proofs are the great 'leveling-up' of mathematics. At my former university they had a bridge class for students to take when they made the leap from more mechanistic mathematics to proofs. I didn't have the benefit of taking it when I was a math major (to my regret, it was painful to level-up), but it looks like the text they use is publicly available:
The proofs in standard Euclidean high school plane geometry are a perfectly good introduction to how to write proofs. A good next step is just a college course in abstract algebra -- sets, groups, rings, fields, prime numbers, the fundamental theorem of arithmetic, the fundamental theorem of algebra, Galois theory (why with Euclidean construction it is impossible to construct a square with the same area as a given circle), why the square root of 2 is irrational, the Euclidean greatest common divisor algorithm, the Chinese remainder theorem, vector spaces, .... Here there are essentially no prerequisites.
The results proved in an abstract algebra course are often just embarrassingly childishly simple so that getting a solid proof is easy. For the issue of writing style for proofs, just pick that up from the text. E.g., usually say since instead of because. The proofs in abstract algebra and, really, the rest of math, are really essentially the same as in high school plane geometry but just written in a less rigid style.
If want a theorem proving course in calculus, first take calculus where theorem proving is not the main content, then take abstract algebra (where will use no calculus), and then take a theorem proving course in calculus, e.g., from W. Rudin, Principles of Mathematical Analysis. With this sequence, should never be without prerequisites or in doubt about the reason, point, value, or intuitive view of the material.
Yes, the Wiles quote is really good for describing research, but learning need be nowhere nearly that challenging.
http://www2.math.umd.edu/~shalper/text.pdf