My question is simply: how does one think about sections of nontrivial vector bundles on a smooth manifold, for example? The canonical example I think of is a vector field, i.e. a section of the tangent bundle, or even a differential form (section of the cotangent bundle; slightly harder to visualize as literally as tangent vectors, but still somewhat concrete). I know that the Mobius strip gives a nontrivial bundle over the circle, which is perhaps the only one I can see clearly because of the low dimension, and even then the sections seem not so intuitive. However, when it comes to an arbitrary vector bundle, I have no idea how to imagine the sections.
My motivation is that I'm trying to learn algebraic geometry and sheaf theory, which is explained to me as trying to generalize vector bundles (by just declaring the sections on an open set), but I'm having trouble how to think of these if they're not just trying to emulate vector fields/differential forms (or of course, "smooth functions" on a space in the case of the trivial bundle, which I like as motivation for the structure sheaf on a scheme). Any examples from (beginner) algebraic geometry would be appreciated too.