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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:

1. $H^0$ of a sheaf coincides with its global sections.
2. Short exact sequences of sheaves give rise to long exact sequences of sheaf cohomology.
3. 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.

People have mentioned several good references, which you would need to consult for proofs of the above. Actually, I'm not sure any of the ones mentioned discuss Dolbeault, so let me add Griffiths and Harris, Voisin, and Wells books on algebraic geometry/complex manifolds. Although these may not satisfy the "not too hard core" requirement.

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:

1. $H^0$ of a sheaf coincides with its global sections.
2. Short exact sequences of sheaves give rise to long exact sequences of sheaf cohomology.
3. 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...

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 the properties:

1. $H^0$ of a sheaf coincides with its global sections.
2. Short exact sequences of sheaves give rise to long exact sequences of sheaf cohomology.
3. 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.

People have mentioned several good references, which you would need to consult for proofs of the above.

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:

1. $H^0$ of a sheaf coincides with its global sections.
2. Short exact sequences of sheaves give rise to long exact sequences of sheaf cohomology.
3. 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.

People have mentioned several good references, which you would need to consult for proofs of the above. Actually, I'm not sure any of the ones mentioned discuss Dolbeault, so let me add Griffiths and Harris, Voisin, and Wells books on algebraic geometry/complex manifolds. Although these may not satisfy the "not too hard core" requirement.

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 the properties:

1. $H^0$ of a sheaf coincides with its global sections.
2. Short exact sequences of sheaves give rise to long exact sequences of sheaf cohomology.
3. It can be computed but 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 $$H^i(\pi_1(X), A) \to H^i(X, A)$$ which is an isomorphism if $X$ is aspherical (Eilenberg-Maclane) but not in general.

People have mentioned several good references, which you would need to consult for proofs of the above.

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 the properties:

1. $H^0$ of a sheaf coincides with its global sections.
2. Short exact sequences of sheaves give rise to long exact sequences of sheaf cohomology.
3. 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.

People have mentioned several good references, which you would need to consult for proofs of the above.