Can anyone please recommend some good reading on the geometry of linear groups and their actions?

An example of the kind of question I am interested in: Explicitly describe a fundamental domain for the action of $GL_2(\mathbb{Z})$ on $GL_2(\mathbb{R})$, and compute the volume of the quotient.

I'm familiar with this particular question and its answer, but it is evidently a special case of a more general theory and I would love to see it treated in context. I looked briefly at Borel's Linear Algebraic Groups, Lang's $SL_2(\mathbb{R})$, a couple intro books on Lie groups -- and at a brief glance, none of them seemed to squarely address this kind of question.

Thank you!