Field's Medal? Is that high enough?

What do you think mathematicians do?

I meant in terms of mathematical field/topic. Like the progression I did to get an engineering BS was roughly

* basic math > algebra/trig > calculus, linear algebra, differential equations

As a non-mathematician, my impression has been that mathematicians do a lot of proof-based work since (to my impression) the goal is to make new findings, not just learn concepts and formulas that they can apply to their work like an engineer. From my limited experience with proof-based work, it seems like something where you would need guidance and discussion. Whereas when I learned things like derivatives and integrals, it was very procedural and you just needed to learn a process, which is then built on by a more advanced process, etc.

Can one realistically self-learn up to low-dimensional topology or algebraic geometry just by reading books and doing the exercises?

The likelihood that you will succeed in self-learning any field of mathematics without outside help goes down as the material gets more advanced, but there's nothing fundamentally impossible about the endeavor. You can definitely get to graduate level that way if you're willing to put in the work and study the right things in the right order.

Luckily, there are plenty of resources available to someone self-studying online on various math forums.

what would you class as "graduate level" in this context? Would that be stuff like PDEs? As I said I'm not very familiar with what the 'progression' is past undergrad