For finite abelian groups, see Section V.6 in Brown's book on group cohomology. For the finite simple groups, the wikipedia page has pretty good information.
As far as a general reference, aside from general books on group cohomology (like Brown's book above, which along with Adem-Milgram's book is my favorite reference) I have found Karpilovsky's book "The Schur multiplier" useful.
I don't know what you mean by the "long exact sequence" in cohomology. What does exist is the Hochschild-Serre spectral sequence. The hardest part of analyzing this is computing the differentials. For computing $H^2$, I have found Huebschmann's papers "Automorphisms of group extensions and differentials in the Lyndon-Hochschild-Serre spectral sequence" and "Group extensions, crossed pairs and an eight term exact sequence" helpful.
There is a five-term exact sequence in group cohomology arising from a short exact sequence. However, this is only useful for computing $H^2$ of the cokernel from information about $H^2$ of the central term and $H^1$ of the kernel. See Brown's book for a discussion of this.