If you're motivated / curious enough / have enough time, I recommend Spivak's Calculus as a way of learning calculus the, as you say, 'proper (whatever that means) way.' It's basically a treatment of calculus from the perspective of real analysis, having the 'no ad-hoc rule teaching' policy at its foundation. You prove things, build up a non-fragmented edifice slowly, and end up being introduced to analysis such that you can then pick up other things (it's a rather extensive treatment of calculus indeed.) Or so goes my narrative in the midst of frustrations regarding self-motivating to continue individually progressing through the book. :)

The 3rd and maybe the 4th editions can be found online by doing an internet search for pdf/djvu files.