What is the best sequence in which to study undergraduate mathematics?
I am someone who did just as you are hoping to do -- I don't enjoy studying material without a strong background in the assumed prerequisites, so some would feel that I'm suggesting too much.
My suggestion is to go through Stewart's calculus front-to-back, doing typical problem sets. Whenever you struggle with a pre-calculus or algebra fact, make sure you review that material. Once you've done this, I recommend deviating from the typical routine and picking up a copy of the book "How to Prove it." With the maturity you have after going through calculus, you will understand the essence of what a proof is, and the importance of working with and thinking in terms of definitions. After this, for fun you might enjoy going through "Geometry: Seeing, Doing, Understanding" by Jacobs. Then, quickly, go back through your Algebra/Pre-Calculus textbook examples and work through any of them that do not seem immediately obvious how to do. Then, quickly go back through Stewart, doing the same thing. You will be very prepared at this point to do very well in your future studies.
From here, you would want Linear Algebra, Complex Analysis, Abstract Algebra, and Real Analysis. For Linear Algebra, I recommend Gilbert Strang's MIT opencourseware videos, and for Real Analysis, I recommend Francis Su on YouTube (he goes through the textbook Principles of Real Analysis by Rudin).
These, I think are essential. you can't plan too much further in advance because your interests will change as time goes on. But some other areas you could look at are probability and statistics, number theory, numerical analysis, and differential equations, or more advanced topics in what I've already mentioned.
Having self-studied (and continuing to study) a lot of undergraduate mathematics, I will tell you the order in which I did things (the list might be a bit ambitious, especially the jump from Apostol's Calculus to Artin's Algebra...you may want to give yourself a bit of background in linear algebra before trying to tackle Artin):
- Mathematical Logic
- Set Theory (introduction to axiomatic set theory)
- Apostol's Calculus Vol. 1
- Multivariable Calculus (used OpenCourseWare)
- Artin's Algebra
- Rudin's Principles of Mathematical Analysis
- Calculus on Manifolds (Spivak+Munkres)
- Topology (Munkres)
If I were to do it over again, I might put less emphasis on mathematical logic and set theory as I did (but that was what motivated me to self-study in the first place).
I would recommend taking linear algebra before studying differential equations--a lot of differential equations exploits linearity of the derivative, and this can be rephrased easily in terms of linear operators.
I would recommend these to be read at the same time:
- Euclid's ELEMENTS (translated by Heath), and
- Hartshorne's GEOMETRY: EUCLID AND BEYOND.
Proving theorems by construction is a wonderful way to do mathematics, and Hartshorne's book will point out and correct the logical mistakes Euclid made, extend his results, and provide a connection to contemporary mathematics by way of introductory modern algebra. Best of all, the prerequisites are zero, zilch, nada. By the time you have finished both of these books (which should take the better part of two semesters going it alone) you will know what you want to know more about.