How can a undergraduate student start learning geometry? I have finished all undergraduate course on analysis and abstract algebra. However, I know little about modern geometry, such as riemann geometry, conformal geometry, and general relativity.
Where should I start my self study of geometry?
 A: Textbooks are of course fine, but I would recommend Anton Petrunin's list of geometry problems.  It covers most of modern geometry and, as he writes in the introduction, it's about cool ideas and problem solving rather than theory and machinery.  Geometry is notorious for compelling people to learn 2 years' worth of theory before you get to do something cool, and Petrunin's problems are an antidote.
Of course, the problems are not at all easy - I'm not sure I would have been able to solve any of them on my own as an undergraduate.  But you can still learn a lot from them as follows:
1) Read the preamble to each section, and consult whatever external references you need to understand the example problem (filling in all missing details).
2) Pick one or two problems according to your interests, and consult whatever references you need to fully understand the statement.
3) Spend at least a few days trying to solve the problem(s), working out a concrete example or two if you can't get anywhere.
4) If you haven't solved the problem after a few days, read the partial solution and then consult whatever references you need to fill in all of the details.
If you do this with all of the problems then you will be better equipped than many people with advanced degrees to do research in geometry (myself included, if I'm being honest).
A: Your question is very general, but I'll share one of my favorites. The first six chapters of Visual Complex Analysis by Needham are an excellent introduction to Non-Euclidean geometry and the associated conformal models. It was here that I first learned the group of Möbius transformations is isomorphic to the Lorenz group from special relativity. 
It is completely approachable at an undergraduate level and will give a nice base to head in various "modern" directions. It had a big effect on me.
You probably also want to look at answers to related questions, e.g. Where to start learning Differential Geometry/Differential Topology? 
