I am teaching a graduate seminar in equivariant topology. The format of the course is that I will give 2-3 weeks of background lectures, then each week a student will present a topic. The students have all taken a basic course in algebraic topology (they know homology/cohomology and fundamental groups), but some may not know much more topology than that. Topics will likely include equivariant cohomology, (equivariant) bundles and characteristic classes, equivariant K-theory, and important classes of examples interspersed, including toric varieties, homogeneous spaces, and the Hilbert scheme of points in $\mathbb{C}^2$. My personal goal is to learn a bit about Bredon cohomology for compact, connected Lie groups (I'm happy to restrict that a bit, but probably not to finite groups).
Some references that I already have in mind include those listed in David Speyer's question and answer about equivariant K-theory. For Bredon cohomology, there are two books: Equivariant Cohomology Theories by G. Bredon, and Equivariant Homotopy and Cohomology Theory by J.P. May (with many other contributors).
Reference request: What are classic papers in equivariant topology that every student should read?
Best Answer
If I may be so bold, I would actually strongly suggest you start with finite groups, rather than compact Lie. While many of the results in equivariant homotopy are true in both cases, the formulations for finite groups are often easier to understand. Additionally, there are twists that show up in the compact Lie case which just make exposition (and I find comprehension) a good bit trickier.
For a finite group, it is very easy to carry out computations with Bredon homology and cohomology. In fact, it's easy to write down chain complexes of Mackey functors which do everything for you. For compact Lie, you can of course, do the same thing; I personally find it substantially harder and less intuitive.