This is a textbook on Borel equivariant cohomology. Since you like homotopy theory, let me phrase it in that language.
Borel equivariant homotopy theory is the homotopy theory presented by G-spaces where the weak equivalences are the continuous equivariant maps which are non-equivariant weak equivalences. In particular, $X \times EG \to X$ is always a Borel equivalence.
Borel cohomology is the standard cohomology theoy in Borel equivariant homotopy theory. It has historically been very useful in practice to geometers, and the textbook you look at has a long discussion of the de Rham model.
Algebraic topologists tend to be more interested in what they sometimes call the "genuine equivariant homotopy category". (I am assured nobody is turning up their noses while they say this.) There, they want the weak equivalences to be such that $f^H: X^H \to Y^H$ is a weak homotopy equivalence for all subgroups $H$. Borel cohomology is too weak to study this, and one studies things called Mackey functors and Bredon cohomology and so on.
You won't learn anything about Bredon cohomology in Tu's book because it's not relevant to his interests. Bredon's notes on the subject are probably still the standard introduction.
To be honest, equivariant topology is a Kafkaesque zoo. It will be difficult to navigate it without an advisor to guide you. If you want a full picture you will need to know the Borel theory as well as the Bredon theory, reading both the geometers and the algebraic topologists. There is a WIP book project by Hill, Hopkins, and Ravenel which might be of interest to you.
