I'm curious how one can classify the bundles over a given manifold. I recently read this paper on classifying $2$-sphere bundles over compact surfaces. A lot of the concepts went over my head since I'm still having trouble doing more "trivial" classifications.
To get used to harder classifications, I'm hoping to get used to doing easier ones first. Thus, I'm curious how one can classify the $2$-plane bundles over $S^2$. To be honest, I'm not sure how to even approach such a problem.