Let me make a few comments here starting with sheaf cohomology. It can be defined in several ways, but the definitions are less important the properties:

- $H^0$ of a sheaf coincides with its global sections.
- Short exact sequences of sheaves give rise to long exact sequences of sheaf cohomology.
- It can be computed by acyclic resolutions, such as fine resolutions on a paracompact Hausdorff space.

Using 3 plus the Poincaré lemma, we get a version of de Rham's theorem that sheaf cohomology of a manifold with coefficients in the constant sheaf $\mathbb{R}$ coincides with de Rham cohomology. A similar application of 3 shows that on a complex manifold, sheaf cohomology of the sheaf of holomorphic $p$-forms coincides with Dolbeault cohomology. Also, on a sufficiently nice space (e.g. a manifold) singular cohomology with coefficients in an abelian group $A$ coincides with sheaf cohomology with coefficients in $A$. This is also true if $A$
is replaced by a local system. A local system comes with a monodromy representation $\pi_1(X)\to Aut(A)$. There is map from group cohomology to singular cohomology
$$H^i(\pi_1(X), A) \to H^i(X, A)$$
which is an isomorphism if $X$ is aspherical (Eilenberg-Maclane) but not in general
(e.g. take $X=S^n, n>1$).

**Postscript**
A more genuine answer, which would be even more unhelpful to the OP than the one I gave, is
that I doubt there is a single coherent and easy treatment of all of these disparate topics.
But there are plenty of good references (some of which have been mentioned) that cover
algebraic topology, group cohomology and its applications,
sheaf cohomology and its applications, homological algebra...

verycarefully explained e.g. in Switzer's Algebraic topology but I find the presentation a bit tedious. $\endgroup$ – algori Nov 23 '12 at 1:10twoof the cohomology theories... If a text mentions two of these cohomologies is generally to connect them, because that is the whole point of having two constructions for the same thing! $\endgroup$ – Mariano Suárez-Álvarez Nov 25 '12 at 20:17