Reddit mentions of Elementary Differential Geometry (Springer Undergraduate Mathematics Series)

Axiomatic "non-euclidean geometry" is something that was studied hundreds of years ago but isn't really an area of math that one studies or learns about anymore. Today, "non-euclidean geometry" (like the geometry of spheres or the hyperbolic plane) is part of differential geometry. There are many undergraduate-level books on manifolds and differential geometry, but I've never really looked at these. One you could try is Elementary Differential Geometry by Pressley.

For elementary differential geometry, just calculus and linear algebra should be sufficient. You can use a book like this for that purpose. For more advanced differential geometry, you will need to know topology and analysis and maybe some algebra as well.

The content of the upper level math courses tend to vary depending on the professor and what they feel like teaching on any given year. I took fundamentals of Geometry with prof. Piper a few years ago. We covered most everything in this book (you can read through the index to get a good idea of what the course contained)

http://www.amazon.com/Elementary-Differential-Geometry-Undergraduate-Mathematics/dp/184882890X/ref=sr_1_2?ie=UTF8&qid=1320607881&sr=8-2

We also did a bit with the more computational side of things, representing geometric transformations with quaternions or matrices, did Maple projects, etc.

This is one is the best textbook for self-study I've find: Elementary Differential Geometry - A.N. Pressley.
Every self-study book should be like this one, well written and with answers to every exercises.