That story really makes me wonder if my understanding of computers would be deeper, or more complete in some way, if I'd had that sort of background/experience...
Take a course on compilers. It will teach you these things and give you a much more fundamental understanding.
Seconded! Also play with assembly language for a small CPU, even in an emulator. Doing assembly in the 68k and Z80 really helped me grok computers. Even the old assembly makes super CPUs of today more understandable.