Shapiro's lemma and dimension-shifting. Cohomology of cyclic groups and Herbrand quotients.

It would be helpful what you need to know group cohomology for.

If you have an interest in pro-p or profinite groups, there's a slew of things to add on here (notably, the interpretation of the ranks of $H^1(G,F_p)$ and $H^2(G,F_p)$ as cardinalities of minimal generator and relator sets, the value of the cup product and Massey products in determining the structure of these relations, the relation with the Schur Multiplicator given by Hopf's formula above). Similarly, if you're interested in group cohomology as Galois cohomology, there's a whole field of mathematics to add on to the list. In addition to class field theory via group cohomology a la Tate, there are a few papers (I remember a particularly good one by Cornell and Rosen) that derive a large portion of a semester of algebraic number theory starting from elementary group cohomology.

Shapiro's lemma and dimension-shifting. Cohomology of cyclic groups and Herbrand quotients.

It would be helpful to know what you need to know group cohomology for.

If you have an interest in pro-p or profinite groups, there's a slew of things to add on here (notably, the interpretation of the ranks of $H^1(G,F_p)$ and $H^2(G,F_p)$ as cardinalities of minimal generator and relator sets, the value of the cup product and Massey products in determining the structure of these relations, the relation with the Schur Multiplicator given by Hopf's formula above). Similarly, if you're interested in group cohomology as Galois cohomology, there's a whole field of mathematics to add on to the list. In addition to class field theory via group cohomology a la Tate, there are a few papers (I remember a particularly good one by Cornell and Rosen) that derive a large portion of a semester of algebraic number theory starting from elementary group cohomology.