There is a kind of "hierarchy of math" known to logicians. This paper calls it the "Godel hierarchy", but a more generic name is just "the consistency strength hierarchy."

Roughly speaking, there is a linear hierarchy of systems ranging from very weak (systems that can only prove very basic facts about arithmetic) to very strong (systems of set theory that prove way more than anything that is needed by "ordinary" mathematicians.) Given some theorem from analysis, abstract algebra, graph theory, topology, etc., we can ask: how far up the hierarchy do you have to go to prove this theorem? Pretty much any mathematical theorem you encounter in, say, an undergraduate math curriculum, can be placed somewhere on the hierarchy. (There are some examples in the link I provided.)

Having said that: the hierarchy has nothing to do with the order in which people learn math. Nor is it really a classification of mathematics in terms of "subjects" like graph theory, combinatorics, etc. The hierarchy has more to do analyzing what kinds of assumptions are needed to prove various theorems.