Not an answer as such, but some extended comments with explicit references (posted as community-wiki).
As a non-specialist, I'm unsure whether you are asking for new information or for better references than you have. Compact Lie groups have been studied thoroughly for more than a century, going back (as Robert Bryant points out) to foundational work by E. Cartan, followed soon by Weyl and others. By now there are many textbook and lecture treatments of compact groups in the context of more general semisimple or reductive Lie groups, along with a few more targeted accounts of the compact groups by themselves: structure, classification, representations, topology. So it's difficult to find much new to say, though not impossible.
Concerning more recent references, I'd mention the Springer GTM 98 Representations of Compact Lie Groups by Brocker and tom Dieck, which lays out the basic theory with detailed examples. For instance, they deal explicitly with central functions in their Chapter IV, followed by more on root systems and representations. Another source is Bourbaki's Chapter IX on compact Lie groups, where the emphasis is often quite analytic and includes a treatment in section 8.3 of Fourier transforms of central functions.
Allen has provided a basic summary in terms of maximal tori and Weyl groups. The sources I've mentioned provide a lot of details about all of this, while the topology (Stiefel, Bott-Samelson, ... ) has been exposed in lecture notes by Bott. It's mostly a question of putting your own concerns in clear focus relative to all this literature. The drawback of Lie theory is the multitude of approaches taken to it, but that also reflects its importance.
I'd mention finally that there is a remarkable algebraic parallel for all these questions about class functions in the theory of semisimple algebraic groups. Although "tori" become algebraic tori, the Weyl groupa and root systems play much the same role. Here the Chevalley restriction to a maximal torus leads again to study of the Weyl group orbits on a fixed maximal torus. In this algebraic setting, the resulting geometry is that of an affine algebraic variety, which in the simply connected case is just affine space. Moreover, the characters of irreducible representations and "characters" of maximal tori come into play in a way completely parallel to the study of compact Lie groups (even without analysis being involved).