Real analysis for a non-mathematician.
A rigorous calculus textbook would probably help you "get into" Rudin. An example is Lang. Spivak is also much liked, and there's Hardy's Course of Pure Mathematics whose first three editions are free and have lots of amusing exercises. However, Rudin has stuff that isn't in Lang or Hardy (or I assume Spivak) in chapters 9, 10, and 11.
As for whether real analysis is a prerequisite for differential geometry, I'm afraid that you need a mastery of the subject matter of Rudin chapter 9 (multidimensional derivatives, inverse functions, implicit functions) for modern differential geometry texts to make sense to you (e.g., Lee's Smooth Manifolds). An alternative might be a style of learning in which there is a separate course in 2- and 3-dimensional manifolds first; for this a common text is Manfredo do Carmo's Differential Geometry of Curves and Surfaces.
Independently of all the above, for math learning you need to (i) learn theorem-proof style and (ii) work exercises, writing down the proofs in "rigorous" mathematician style.
Given what you've said, for real analysis I recommend looking at these three:
Yet Another Introduction to Analysis by Victor Bryant
Bryant's book is excellent for reading without an instructor. All the exercises have solutions. The book is full of pictures, diagrams, and other pedagogical aids. The book is excellent as a direct sequel to the standard calculus sequence and it is carefully written with this in mind.
Advanced Real Calculus by Kenneth S. Miller
Miller's book is notable for its no-nonsense straightforward and uncluttered style. Miller, also the author of Engineering Mathematics (which was later reprinted by Dover publications), got his B.S. in chemical engineering before going on to get M.A. and Ph.D. degrees in math. Because of his background (Miller also taught engineers for many years and has worked in both government and industry as an applied mathematician), Miller's book is definitely worth considering. As a mathematician he's not a "Rudin", but for what's it's worth he did do a post-doc at the Institute for Advanced Study in Princeton.
Schaum's Outline of Theory and Problems of Real Variables by Murray R. Spiegel
Although Spiegel's book gets into Lebesgue integration (but only for the reals), much of it should be accessible to you, especially for supplementary reading and for the worked problems. Spiegel, who you may recognize because you are an engineering student, is the author of a lot of Schaum's Outline books on engineering math topics (complex variables, Laplace transform, vector analysis, theoretical mechanics, difference equations, etc.), and his real variables book reflects this by including a chapter on Fourier series.
I don't think differential geometry is a good choice for "expanding your horizons", but if you really want to do this, I think Elements of Differential Geometry by Richard S. Millman and George D. Parker would be a far better choice than Manfredo do Carmo's Differential Geometry of Curves and Surfaces that someone else mentioned. FYI, I've taken a course that used do Carmo's book and I've audited a course that used Millman/Parker's book.
As for abstract mathematics, you might want to look at Paul Roman's 2-volume work Some Modern Mathematics for Physicists and Other Outsiders Volume 1 contents. Despite the wide selection of advanced topics covered, these books do a nice job of making things accessible to someone not steeped in Bourbaki style mathematics.
(ADDED 4 DAYS LATER) The morning after I wrote the comments above, I looked at my copy of Schaum's Outline of Theory and Problems of Real Variables and realized I made a mistake in recommending it. I thought most of the book was pre-Lebesgue integration topics, with Lebesgue integration only introduced in the later parts of the book. In fact, the book begins with Lebesgue measure and integration, and thus it would not be appropriate for someone who has not studied undergraduate real analysis or advanced calculus. Aside from the mismatch with your needs, the book is fine.
As it was the weekend, and I was sick and couldn't do much else anyway, I decided to carefully look through the real analysis books I have (over 100 texts at the mid undergraduate level, advanced undergraduate level, and beginning graduate level) and see what I could find that might be worth suggesting besides those I've mentioned.
In doing this, I only considered relatively short texts that could be covered in one semester, because when self-studying lengthy texts it is very easy to get overwhelmed and then procrastinate and not accomplish much. For the most part, I also only considered texts that were not especially abstract or formal, so for example if metric spaces appeared at all, they played a very minor role overall, and formalities such as axiomatic set theory and Peano axioms did not intrude into the introductory material. Finally, I looked for books that seemed to have topics or emphasis that might be useful to engineering, such as Fourier series.
I found two additional books that best fit these conditions, one by Labarre and one by Randol. My personal favorite for what I believe would best work for you is Randol's book, and this includes those books I have already mentioned.
Because it is difficult to find anything specific about older texts on the internet (no one seems to go to the trouble), I've written some comments about each book. I've also included some more information about Kenneth S. Miller's book, for the same reason.
Anthony Edward Labarre, Intermediate Mathematical Analysis, Holt, Rinehart and Winston, 1968, xvi + 253 pages.
Very nice treatment with a more than average coverage (for one semester length books) of topics -- complete ordered fields, cardinal numbers, equivalence of sequential and neighborhood limit notions, structure of open sets in the reals, uniform continuity, derivatives have the Darboux property, contraction mappings, the measure zero characterization of Riemann integrability, uniform convergence of sequences of functions, Picard's theorem on the existence of a unique solution to $\frac{dy}{dx} = f(x,y)$ and $y(x_0) = y_0$ where $f$ is continuous and Lipschitz with respect to $y$ in some rectangle containing $(x_{0},y_{0})$ in its interior [proved using contraction mappings], final chapter (pp. 202-237) on Fourier series that includes discussions of the Dirichlet Problem and Fejér's Theorem. Main Weakness: There is very little material on the convergence and divergence of infinite series of real numbers, and there is almost no material on power series.
Burton Swank Randol, An Introduction to Real Analysis, Harbrace College Mathematics Series, Harcourt, Brace and World, 1969, xii + 112 pages.
Randol's book seems to be primarily focused on the convergence and divergence of series of functions. Thus, there is little about the reals as a complete ordered field, little about the structure of open sets of reals, little about the Darboux property of derivatives, etc. On the other hand, Randol includes such topics as: Cauchy's theorem on absolute convergence of double series, all of Chapter 3 (out of 6 chapters) is devoted to power series (including a discussion of the power series expansion of $f(g(x))$ in terms of the power series expansions of $f(x)$ and $g(x),$ and a section devoted entirely to the existence of an everywhere $C^{\infty}$ function that is not real-analytic at $x=0),$ all of Chapter 4 is devoted to the Weierstrass approximation theorem for continuous functions by polynomials (the Karamata tauberian theorem is given as an application), all of Chapter 5 is devoted to Fourier series (a final section proves as an application that for any irrational number $\alpha,$ the sequence $\alpha,$ $2\alpha,$ $3\alpha,$ $\ldots$ is equidistributed $\mod 1).$ The last chapter is devoted to a brief introduction to the Lebesgue integral.
Kenneth Sielke Miller, Advanced Real Calculus, Harper's Mathematics Series, Harper and Brothers, 1957, x + 185 pages.
Contents: 1. Numbers and Convergence (pp. 1-19). 2. Topological Preliminaries (pp. 20-25). 3. Limits, Continuity, and Differentiability (pp. 26-47). 4. The Riemann Integral (pp. 48-84). 5. Sequences and Series (pp. 85-122). 6. Functions of Several Real Variables (pp. 123-139). 7. Integrals of Functions of Several Variables (pp. 140-165). References (p. 166). Appendix. Solutions to Some Exercises (pp. 167-182). Index (pp. 183-185). Main Weakness: There is very little topological material in the book, this being sacrificed for the book's extensive treatment of epsilon-delta arguments. From the Preface: We have attempted to present the subject matter of real variables at a rigorous and logically complete mathematical level and at the same time not make the tome so ponderous that the beginning student is buried under a mass of notations, definitions, and peripheral material. $[\ldots]$ Keeping in mind the class of readers to whom the text is addressed, the majority of proofs are carried out with painstaking detail--no epsilon or delta slips by unnoticed. Note: In a curious bit of old-fashioned terminology (even for 1957, at least in mathematics texts although perhaps not in physics or engineering texts at this time), Miller talks about single-valued functions on p. 26, saying that only single-valued functions will be considered in the text, and he gives $f(x) = \sqrt {x}$ as an example of a function that is not single-valued. This is all the more curious because on the very next page (p. 27), when proving by epsilon-delta methods that $f(x) = x^2$ is continuous everywhere, Miller writes "Let $\delta = \min \left( \sqrt{\frac{\epsilon}{2}}, \, \frac{\epsilon}{8} \right),$ then $[\ldots],$" and in this usage he is obviously using the symbol $\sqrt{\,}$ in its current single-valued sense.
Check out Understanding Analysis by Abbott. The title says it all.
Ahem ... I hear that if you go to a certain website to search for the instructor's solution manual for this book, you can find a .pdf version that someone accidentally left lying around for anyone to find. I understand that it contains a complete copy of the text, followed by complete solutions -- very useful for self-study.