Wanting to learn mathematics from "first principles" brought a lot of comments from graduate-level mathematicians. While their advice applies very much for mathematics students, I can't recommend going down that road for engineering types.
In mathematics, everything is connected. One can build up a specific topic from first principles only. But with a too narrow focus one looses these lovely connections between different fields that allow to change the perspective on how we think about problems.
I was in a similar situation some 2 years ago. Try "Vector Calculus, Linear Algebra, and Differential Forms" by Hubbard and Hubbard. You will not be disappointed. Yes, you get (enough) rigor and a lot of first principles mathematics. Nonetheless, the authors have found a lovely way to integrate a wealth of important results from many fields into a coherent text that has one goals: letting you understand the connections and letting you solve the problems.
Hubbard & Hubbard is a fantastic book; for the right audience, it may be all that is needed. I'd say that every mathematically minded engineer or physicist certainly should have it. It does a fantastic job of exposing the fundamental ideas of linear algebra and multivariable calculus--and in a way that doesn't require really any prerequisites beyond basic calculus. Reading this book is like being in a lecture with an experienced practitioner of both pure and applied mathematics simultaneously (well, this is actually true of Hubbard, but most texts are not written in a way that the author is so detailed and clear so as to seem present); the notes in the margins and the extensive, direct, and clear explanations are absolutely lovely.
That said, somebody interested in building a foundation for pure mathematics, and not so much motivated by the ability to solve problems outside of mathematics, would probably be better served by reading a standard text on real analysis.
On the other hand, I myself have turned to the notes in the margins of Hubbard & Hubbard, even when studying real analysis from the purest point-of-view, because the little tidbits are just so insightful.
Somebody who really took Hubbard & Hubbard seriously, though, could come away with a monster understanding of applied mathematics, while still having learned the craft in a way that is correct enough to lead to further study in pure mathematics as well. Nobody can really go wrong having this book on his or her shelf (although it is a bit expensive).
As a mathematics textbook it of course provides rigorous proofs. But it doesn't neglect to develop intuitive insight (that can be less rigorous yet still very effective).
Here's a quote given in chapter two. That's some motivation to develop more mathematical maturity as an engineer!
In 1985, John Hubbard was asked to testify before the
Committee on Science and Technology of the U.S. House of
Representatives. He was preceded by a chemist from DuPont,
who spoke of modeling molecules, and by an official from the
geophysics institute of California, who spoke of exploring
for oil and attempting to predict tsunamis. When it was his
turn, he explained that when chemists model molecules, they
are solving Schrödinger’s equation, that exploring for oil
requires solving the Gelfand-Levitan equation, and that
predicting tsunamis means solving the Navier-Stokes equation.
Astounded, the chairman of the committee interrupted him and
turned to the previous speakers. “Is that true, what
Professor Hubbard says?” he demanded. “Is it true that what
you do is _solve equations_?”
In mathematics, everything is connected. One can build up a specific topic from first principles only. But with a too narrow focus one looses these lovely connections between different fields that allow to change the perspective on how we think about problems.
I was in a similar situation some 2 years ago. Try "Vector Calculus, Linear Algebra, and Differential Forms" by Hubbard and Hubbard. You will not be disappointed. Yes, you get (enough) rigor and a lot of first principles mathematics. Nonetheless, the authors have found a lovely way to integrate a wealth of important results from many fields into a coherent text that has one goals: letting you understand the connections and letting you solve the problems.