RE: Clifford Algebras.

A great introductory book is Geometric Algebra for Computer Science:

http://www.geometricalgebra.net/index.html

This book includes a lot of the geometry side of Geometric Algebra.

For a more axiomatic treatment, see Doran & Lasenby's "Geometric Algebra for Physicists":

http://www.mrao.cam.ac.uk/~cjld1/pages/book.htm

Note that the applications in this book require a great deal of physics background, and aren't for the faint of heart.

I'm continually impressed how many different aspects of mathematics can be brought under the umbrella of GA. For example, all of Stokes theorem, Green's theorem, Cauchy integral equations, and Divergence theorems are all specific cases/applications of the Fundamental theorem of geometric calculus. There are a number of other algebraic structures that have GA representations. Examples include: Cramer's rule, differential forms, complex numbers, quaternions, and Pauli and Dirac algebras.