TITLE: Self Teaching Analysis, Topology and Differential Geometry
QUESTION [5 upvotes]: I posted some questions similar to this one not long ago but I think I phrased them wrong and as such got valuable input but not really an answer to my questions. If reposting a similar question like this is against the rules of this forum please tell me! I am still new to this site and eager to learn.
Basically I am in electrical engineering (3rd year) but I think I should have done an undergrad in math. I have a limited interest in application and am really much more interested in class when we do rigorous math. At this point I am considering either doing a masters degree in math or going into Control Theory (not control systems, but rather the mathematical theories behind nonlinear control which I guess is more applied math than engineering).
What I want to do now is go through textbooks on my own to make up for the butchery of math that happened in my engineering classes. The great thing is that I will have a year-long internship starting in May during which I will have time to dedicate to this. I want to emphasize that although there is a strong chance I will go into Control Theory, I want to cover rigorous math and detach myself almost entirely from engineering.
My real question/concern is what textbooks to use for each subject and what order to do it all in. Here is my current plan, I would very much like your input:

Real Analysis by Chapman Pugh (I'm already quite deep into it and loving it)
Topology by Munkres (Part I: General Topology)
Abstract Algebra by Dummit and Foote (Group Theory)
Topology by Munkres (Part II: Algebraic Topology)
Smooth Manifolds by John M. Lee

And then perhaps more of the Abstract Algebra textbook and/or an intro to Dynamical Systems, Chaos and Fractals.
Please note I do have some experience in proofs despite engineering. Some of it is due to computer science courses behind quite rigorous and the rest is self practice. For example I am finding Chapman Pugh quite accessible.

REPLY [2 votes]: I think that Eric Auld's answer is fantastic, and I agree with most of what he said. I'm adding an answer just to throw in my own two cents. 
Learning some algebra is good, but if you want to do control theory you might be better off initially spending less time on that and more time on learning some dynamics/ODEs. For a first dynamics introduction, "Nonlinear Dynamics and Chaos" by Strogatz is very good and a pleasure to read. For ODEs, I'm a fan of "Differential Equations, Dynamical Systems, and Linear Algebra" by Hirsch/Smale (the 1974 edition), and also the ODE text by V.I. Arnold. 
Chapters 1-3 of Munkres is a good first exposure to topology but I think that Lee's topological manifolds book is especially useful background for differential topology/geometry. I also liked the algebraic topology introduction in Lee's book a little better than the one in Munkres. 
Lee's smooth manifolds book is now my favorite on the subject, but I read other books first. You might also like the books by Boothby and Guillemin/Pollack.
I was in your (almost) exact same shoes 5 years ago: third year electrical engineering student, unhappy with rigor in engineering classes, realized I should have been a math major, reading mathematics books, etc. 
I am now an electrical engineering PhD student in control theory/dynamics. Feel free to PM me if you ever want to talk more!