Interpretation of cohomology of small degree:

$H^1(G,A)$ = crossed homomorphisms $G\to A$ modulo principal ones.

$H^2(G,A)$ = equivalence classes of extensions of G by A.

$H^3(G,Center(G))$ = obstructions to existence of extensions of G by A.

2. Transfer and its applications: If $G$ is finite then

1. $H^i(G,M)$ is a torsion group annihilated by multiplication by $|G|$.

2. Embedding of $p$-primary component of $H^i(G,M)$ into a subgroup of $H^i(P,M)$, for any $p$-Sylow subgroup $P\subset G$.

In general, Brown's book "Cohomology of groups" gives a decent overview of what is good to know.