The general theorems about etale cohomology are usually enough to let us compute a given $\mathrm{H}^i(X,\mathbf{Q}_\ell)$ as a $\mathbf{Q}_\ell$-vector space without too much difficulty.
I would like to go further and learn how to explicitly compute the Galois action on these groups. But I'm not really sure what the techniques for doing this are. Could someone illustrate how this is done or point me to a reference which does? Are there important base cases from which everything else is built up on?