I am currently following a course which is on analysis on manifolds. The problem is that while I have strong knowledge on topics of analysis (functional, harmonic, etcetera), I know almost nothing regarding the theory of manifolds. This is obviously quite problematic.
Thus far I have read the first two dozen or so pages of Lee's 'Introduction to Smooth Manifolds' and thereby became acquainted with charts, atlases et al. But I would really like to know if there is a fast way for me to acquire this vector bundle language with its "fibres", "sections", "densities", "forms" et al after which I can rely on my intuition?