I am looking for a higher undergraduate or graduate level textbook that introduces group actions after groups just as many textbooks introduce modules after rings.
I think the semigroup/semigroup action vs. group/group action vs. ring/module ("ring action") symmetry is not currently exploited as much as it could be. Especially considering they are mild abstractions of a collection of plain vanilla functions acting on a set.
On the same note, a reference to an abstract algebra textbook with a significant chapter on semigroups (and actions) would be appreciated. A semigroup action is just a few points of data short of a finite automata. Because of this, and because of formal languages, semigroups show up a lot in computer science, and one can get quite far quite quickly. There are also deep algebraic results, see for example the Chomsky–Schützenberger enumeration theorem.