One should read about standard logic, propositional logic, term logic and nth order logics before, because set theory is founded by them.
For example, you can do a lot of math without knowing the definition of real number.
You just need to walk someone through 'sqrt(2) is irrational ' and 'pi is transcendental' and you essentially end up with 'I want all converging sequences in Q to have a limit'. Which gives you the reals.
In some sense, the reals are the most naive number system. Just start with the natural numbers, and allow every sensible opperation.
It is simple only in retrospect. It took a few millennia for mankind to reach the definition of the reals. Meanwhile, Archimedes, Euclid, Pythagoras, al-Kwarizmi, Newton, Fermat, Gauss, Euler, Jacobi, Newton, Galois, Bernoulli, Laplace, Lagrange, Fourier did quite a lot of things (which are very relevant for all fields of math today) without even knowing the definition of limit!
Newton invented limits. Only he did not have a fundamental basis for it. Hence, he saw it is a dirty trick that happened to yield correct and useful results. The formalization of limits came later when the intuitive use of a limit started failing. If I recall, this was with the Weierstrass function.
I think we agree though. You can do most maths just by assuming you can do things with the rational numbers. It takes some really weird stuff before you need to properly define the 'real-numbers'.