Are you looking for an introduction to fiber bundles themselves or to methods of computing the cohomology of the various pieces?
The comments already have some very good recommendations on sources for learning about what a fiber bundle is. As far as their topology goes, I suppose I'll mention two facts:
- The first is about the Euler characteristic and actually follows in some ways from the next point. In fact, one only needs a fibration $p : E \rightarrow B$ here, although some technical conditions are needed: $B$ should be path-connected and the fibration should be orientable over a field. Let $F$ be the fiber of the fibration. Then $\chi(E) = \chi(B)\chi(F)$.
Here's an example of this in practice. Identify $S^3$ with $SU(2)$ and implement the Hopf fibration $S^1 \rightarrow S^3 \rightarrow S^2$. The subgroup $U(1)$ is realized as $S^1$ and the quotient $SU(2)/U(1)$ is realized as $S^2$. This gives rise to a fibration $SU(2) \rightarrow SU(2)/U(1)$ with fiber $U(1)$; hence $\chi(SU(2)) = \chi(S^2)\chi(S^1) = 0$.
(Note that this can be seen also by observing that $SU(2)$ is a compact Lie group of positive dimension; hence it admits a nowhere vanishing vector field and must have Euler characteristic zero.)