Gentle introduction to fibre bundles and gauge connections

for fiber bundles,you may look into novikov's modern geometry part 2. it gives nice explanation and a good place to do learn some "real geometry"

You can find the definition of a fiber bundle and some examples on pp 376-379 of Hatcher's online book Algebraic Topology. You might also consult "Fiber Bundles," chapter 4 of Lecture Notes in Algebraic Topology, by Davis-Kirk. A fast introduction to connections and curvature can be found here. In the case of surfaces, chapter 3 of these lecture notes might be useful to you.

For a 'physicsy' viewpoint, checkout "Geometry of Physics" by Frankel.