then try it after the class is over. take just one theorem from the class and really try to understand it. eventually you will have a few key ideas that you really understand, and everything else will seem like a simple corollary of those. e.g. after decades of teaching studying and writing about it, I can say that all of the structure theory of an advanced linear algebra class, jordan form, rational canonical form, and so on, is a simple consequence of the euclidean algorithm. So if you want to understand the structure of finitely generated modules over Euclidean domains and then pid's, first learn well the euclidean algorithm. then see if you can understand why this is all there is at work in those other theories.
for non commutative algebra, a basic idea is a group acting on a set.
for commutative ring theory, a fundamental result seems to be the noether normalization lemma.
in manifold theory, the basic theorem is the inverse function theorem, and then the implicit function theorem. In many situations, a key result is green's theorem, and then its generalizations, the general stokes theorem.
