What is a really good problem to get your hands dirty in computational-geometry? Computational geometry is an area I find pretty interesting, and I'd like to devote about a month or two to a project that will introduce me to this and help me learn key concepts.

What is a good way to approach this and what are the key concepts I should be sure I'm introduced too?
To mix Suresh V.'s and Dave C.'s suggestions, it might be fun to try to gain experimental evidence
on an unsolved problem by implementing the necessary algorithms.  For example, it is now known that the Delaunay triangulation is not a ($\pi$/2)-spanner
[Prosenjit Bose, Luc Devroye, Maarten Löffler, Jack Snoeyink, Vishal Verma: "The spanning ratio of the Delaunay triangulation is greater than $\pi$/2." CCCG 2009: 165-167.]
You could implement a Delaunay triangulation algorithm, and shortest paths, and try to
determine experimentally what the true spanning ratio might be.
Or, more challenging, try to compute the combinatorial complexity of the Voronoi diagram of lines in $\mathbb{R}^3$,
another unsolved problem (and in the list that Suresh mentions as Problem 3.)