A lot of intuition in math is from geometry. High school plane geometry is a good start, but high school solid geometry, where get good tools and intuition for seeing things in 3D instead of just the 2D of plane geometry, is quite a bit better.
Another part of intuition is from Max Zorn (from Zorn's lemma statement of the axiom of choice):
"Be wise, generalize."
E.g., for the set of real numbers R and a positive integer n, a lot that goes on in the n-dimensional vector space R^n is a generalization of what can see in 3D, e.g., from solid geometry.
E.g., in both cases, a biggie is a perpendicular (orthogonal) projection and, again, the Pythagorean theorem. E.g., regression in statistics is a perpendicular projection.
Perpendicular (orthogonality) is a biggie and is a major part of, say, Fourier series. I.e., each of the sine/cosine waves used is an orthogonal axis, and to find the corresponding Fourier series coefficient just project onto that axis. The projection is an integral of a product, and that is commonly an inner product which close to just a cosine of an angle as in plane and solid geometry and a perpendicular projection and close to correlation in statistics, etc.
E.g., a huge fraction of applied math is from analysis in pure math, and from G. F. Simmons the two pillars of analysis are "continuity and linearity". Linearity generalizes enormously: The quantum mechanics super position is essentially linearity. Under meager assumptions, differentiation and integration in calculus are linear operators. In probability theory, expectation is a linear operator. The wave equation is a linear partial differential equation. Linear programming works on linear equations. Of course, in linear algebra, matrix multiplication is a linear operator. When something is not linear, it may be locally linear which can be enough to get useful results.
For more, a good lesson is to approximate: Commonly we can't get just what we want in just one step but can iterate and approximate as closely as we please. So, can use simple things, sine waves, polynomials, continuous functions, and more, as means of approximation. Such approximation gets us close to more in continuity and, in particular, completeness -- the real numbers are complete and the rational numbers are not but via iteration can approximate the reals as closely as we please. Then this generalizes: The big point about Hilbert space (as mathematicians but not always physicists define it) is completeness. A joke, partly correct, is that "calculus is the elementary consequences of the completeness property of the real number system". E.g., the integral in calculus (and its better version in measure theory) is defined in terms of an iterative approximation. So, if you are good with sine waves, polynomials, continuous functions, wavelets, and more, then you can iterate and approximate a lot, in many cases, everything there is in that case.