I would definitely suggest his two-part paper, The Spectrum of an Equivariant Cohomology Ring.

He develops a ton of information concerning $H_G^*(X)$, and forms the basis for much of its use in future papers.

In terms of studying group cohomology, you should also check out his short paper Cohomology of Finite Groups and Elementary Abelian Subgroups (by Quillen and Venkov) which establishes the celebrated result: If $u\in H^*(G,\mathbb{Z}_p)$ restricts to zero on every elementary abelian $p$-subgroup of $G$ (a finite group), then $u$ is nilpotent. This paper is not even two pages long, although his original proof was contained in a different paper ("A Cohomological Criterion for p-Nilpotence").