I recon that the easiest way to get started with bug hunting might be to just set up afl-fuzzing on some trivial code that has lots of potential for going wrong. Stuff like JSON/HTML/HTTP parsing in C is a great candidate to find integer overflow or buffer overrun bugs in (some of the bugs in this list are exactly that). Throw some CPU time for the fuzzing, and pretty soon you should have a handful of repro cases.
The nice thing about these is that the bugs could be very trivial to fix but have enormous security consequences.
If someone wants a suggestion for a project to try some fuzzing against, the new h2o/libh2o web server and its HTTP parser component (picohttpparser) look very well written but not very well tested (only a handful of hand written, hard coded test cases). I'm pretty sure there's one or more potentially disastrous bugs in it.
edit: almost all bugs in this list seem to be integer overflow bugs. That hints that these issues were found using a static analysis tool like ClangAnalyzer or Coverity.