nLab: Vector Bundle
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Wiki reference entry defining vector bundles in category-theoretic terms, covering transition functions, the correspondence with locally free sheaves, classifying spaces and links to K-theory. Useful for checking precise definitions once the basic differential-geometric picture is familiar.
More resources on Vector Bundles
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Differential Geometry: Connections, Curvature, and Characteristic Classes
Graduate text by Loring Tu developing Riemannian geometry, connections and curvature on vector bundles, and Chern-Weil theory. Readers finish able to compute curvature and construct Pontrjagin, Euler and Chern classes from a connection on a bundle.
nLab
Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.