How could I become a mathematician with only a bachelor's degree in computer science? TL;DR My bachelor's degree didn't leave me with a solid education in mathematics and I'm not sure how to start a career in pure mathematics from there.

To start with, I'm 24 years old, living in Indonesia in Southeast Asia, and I just graduated from a low-ranking university (even for this country) with a bachelor's degree in computer science. I want to pursue a career in mathematics even though I've also enjoyed programming immensely (see all my posts on codegolf.SE) because I've loved mathematics for much longer and I feel like I wasted this first opportunity for education on a less-than-stellar degree. (Note: this less-than-stellar degree didn't even have a linear algebra course. Make of that what you will.)

I figure that a good path to a career in mathematics academia is to at least work towards getting a Master's and PhD in mathematics. The trouble is, is that even this is very broad. What topics/fields should I pursue (first) when looking for graduate programs? If I email professors, what do I tell them? "Hi, I'm looking for a graduate degree, but I don't even have an undergraduate's understanding of mathematics and I have no idea what path to pursue. Can you help me?"

Another hiccup is that I've just started a job as a software developer here in Indonesia to support my family, and I don't think I can leave that job to pursue academia for however many years, until I've worked there for a few years and saved up some money (especially if I pursue these degrees abroad). "Hi, I'd like to join your graduate program, but not till 2023." doesn't seem like a great thing to receive in your email.

This delay does give me time to give myself an informal refresher in undergraduate mathematics, but I have no idea where to even start. Read a textbook? Join an online course? Find a tutor here in Indonesia?

This is not to mention that I'm not sure how on earth I'll pay for everything, or that for most master's or PhD courses (or a combined degree) expect you, quite reasonably, to have a bachelor's degree in mathematics.

And lastly, my main mathematical interests are in number theory, algebra, and calculus, in approximately that order. I was thinking of applying to a North American university (maybe the University of Waterloo in Canada) before applying to Australian or European universities. I have US citizenship, which should help a lot.

In short, I have a myriad of choices and a bad starting position. I understand this is a very general question, so I thank you for any advice you can give or resources you pass along.
Undergraduate math has a wide variety of free or cheap resources online.  Youtube, udemy, and khan academy are your best friend. You will quickly learn how to spot a good teacher / course.  Some books are public and many are available at libraries or online. And of course stackexchange is an option when you feel stuck. As you said, you have a number of years before you would try to make this transition.

This is coming from someone who went from being a teacher to learning python and getting a job at more than double the pay with just 3 months of online learning (mostly udemy and youtube) and a 3 month internship after that

As you learn more and gain a stronger grasp on undergraduate mathematics, you will begin to answer your own question about what topic you really want to pursue.  

Although, even now, it does not hurt to send out a few emails and ask professors about topics.  The worst that can happen is them ignoring you.  You might "waste" 3 minutes of 100 professors time but a single answer could inspire you or help you make a more informed choice about your future careers.  I think it is worth is.

If you are unsure the topics you should study on and the recommended order, find transcripts or graduation requirements for a mathematics degree in a school you respect. It might take a little digging, but finding exactly what you would need to take at x school for y degree is not impossible (heck, send emails to counselors there and just ask if you cant find it easily online)

for an undergraduate reference 

https://webapp4.asu.edu/programs/t5/roadmaps/ASU00/LAMATBS/null/ALL/2018?init=false&amp;nopassive=true

Even vague terms like "upper division depth course" can be clicked on there and it leads to a list with descriptions of what they are and what is taught.

I found this by just googling 

course requirements for math degree ASU

similar searches should yield fairly similar hopefully helpful results