This is an email from Simon King, whose optional Sage package
p_group_cohomology can compute some Group cohomology explicitly when
the group is a p-group.

There is the book of my former boss in Jena, David Green, studying the
computation of cohomology of groups of prime power order,
http://www.springer.com/math/algebra/book/978-3-540-20339-1,
[NOTE: The book below is available for free to UW people
via Springer Link].
D. J. Green:
Grobner bases and the computation of group cohomology.
Lecture Notes in Mathematics, 1828. Springer-Verlag, Berlin, 2003.
In particular, it explains how to compute minimal free resolutions
over the group algebra of a group of prime power order by means of
non-commutative Grobner bases.
Another valuable reference is
D. J. Benson:
Dickson invariants, regularity and computation in group cohomology.
Illinois J. Math. 48 (2004), pp. 171-197.
Available at the arxiv.
It explains how one manages to finish the computation after finite
time (completeness criterion).
Both references are absolutely essential for the spkg. Note that the
spkg currently only covers the case of groups of prime power order. I
am now extending it so that it can compute the modular cohomology
rings of any finite group (ambitious aim: compute the mod-2 cohomology
for all sporadic groups whose Sylow 2-subgroups are of order at most
1024), using the stable element method. I learned it from A. Adem,
J.Milgram: Cohomology of Finite Groups. Springer-Verlag (1994).
The problem with all these references is that they can hardly be
considered "introductory". But perhaps they contain further pointers?