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.