<p>You don’t develop mathematical maturity from calculus courses. To use your analogy, that’s like saying, I’m going to train for the NFL by practicing football with my high school team.</p>
<p>A first course in real analysis is meant to develop your mathematical maturity. You should take it sophomore year at the latest. If you have some proof experience already, then you can start with it freshman year. If not, take a course using Apostle or Spivak your freshman year.</p>
<p>As for the calculus material, if the OP is interested in math/engineering he’s probably done calculus in high school, and you don’t need anything beyond that for real analysis. Sure, there’s vector analysis and differential forms in the second semester of real analysis, but having seen vector calculus beforehand is not needed at all.</p>