The minimal classical way to bootstrap your math knowledge is algebra (mostly linear) + analysis. This is the approach used by e.g. Harvard Math 55, which is a really famous course.
Harvard Math 55 employed Halmos + baby Rudin as main textbooks. Halmos has now been replaced by Axler, which is an excellent textbook (but the typographic changes in the last edition are very distracting).
Rudin is probably too synthetic and dry for a beginner. You could easily replace it by Hubbard & Hubbard (which was the sole main textbook in one Harvard Math 55 edition), or use an aid text like Gelbaum & Olmsted. You can also skip Axler if you go the Hubbard way.
A favorite open question of mine is what would a math bootcamp look like if you went up one level of abstraction and focus more on logic and abstract algebra.
I would love a bootcamp like that. Or even one that helped you with proofs. I've found that's my biggest issue trying to teach myself pure mathematics. I just can't start the damn proof. Once I get the start, I can usually finish it, but starting the proof I just feel so clueless how to do it.
You can learn to do basic proofs the way old school Tsarist and Soviet Russia students did, by studying geometry from Kiselev. It should be pretty straightforward for any adult, and a nice stepping stone to build your knowledge on.
A more advanced approach would be to go through a logic book like Velleman that teaches proofs as structured programming.
This is what they did at my undergraduate university
Do all of these in order first:
Calculus 1 and 2
Linear algebra and multivariable calculus and an introduction to proofs / logic course (you are ready for some electives at this point)
Ordinary differential equations
Any of these can be done concurrently, choose one Analysis and one algebra :
Advanced calculus (eg “understanding analysis” by Abbott)
Linear algebra in the sense of finite dimensional vector spaces
Easier abstract algebra (senior level classes are eg Artin and rudin, these ones are more elementary textbooks)
Core Senior level courses that you take if you want to get good at math:
Analysis sequence (1 year on baby
Rudin)
Algebra sequence (1 year on artin)
Topology (munkres)
Electives:
Probability (can be done after multivariable calc)
Linear optimization (after linear algebra + multivariable calc)
Logic (compactness completeness godel etc whatever, can be done after intro to proofs course but will probably make less sense if you didn’t study some more stuff first)
Numerical analysis (after ODEs I guess or calculus + linear algebra if you want to skip tht stuff)
Statistics (after probability)
Combinatorics - after calc 2 and linear algebra
Geometry - after multivariable calc, linear algebra, proofs
Intro Differential geometry: after advanced calculus
Don’t really have much more knowledge for graduate courses etc. or even some common ones like complex analysis. if you know the senior level core stuff you’re probably “good enough” to make some progress on a lot of things. Each of these classes is 100-200 hours of total study so it seems odd to me that someone will just try to study it on their own by there you go I guess
Not a answer to your question but this query about dependency graph off skills comes up so often that we're trying to build this once and for all at https://github.com/learn-awesome/learn-awesome
It's open-source and a community effort so you're welcome to join & contribute.
Yes, I sketched out textbook suggestions for almost all of OP's topics in a preferred order for reading them. You can roughly segregate material (at the early undergraduate level) into algebra and analysis sequences. See my other comment in this thread.
Is there some dependency order someone could quickly sketch out for some of these topics? Eg, linear algebra comes before X?
HN is an incredibly useful crowdsourcing resource for the self-motivated!