What we have learned today is only the begining of a very fruitful theory. The generalization that the group theory framework provides is so powerful that the advance in understanding some many different parts of mathematics is hard to imagine at first.

One of the major works in mathematics of the 20th century is the classification of all finite (simple) groups. This work is phenomenal and vastly impressive in number in pages it requires to present. It basically means that we now understand any finite set of symmetries! See this article for more information and links about this classification.

That classification involves the representation theory of (finite) groups. A representation is way of connecting an abstract group with the more geometric intuition and properties of objects related to our original idea of symmetries. This notion of "more geometric" realization of abstract algebraic objects extends to other entities like algebras. This excelent article provides a thourough introduction to the subject.

Évariste Galois, a French mathematician from the 19th century, by looking at an uncanny connection between field theory and group theory, managed to solve a very old problem concerning solutions to polynomial equations. Today, this result is part of the beautiful Galois theory.

For a more practical application to group theory and permutation groups, one can read the following notes that present an application of those notions to card tricks.

Concerning set theory and the related paradoxes, one can refer to the following links: A Little Set Theory and The Paradoxes of Set Theory.

Finally, make sure you visit the Cornell Math Explorer's Club for material on other mathematical subjects.