Let's say you want to start contributing to an open source project with thousands LOC. I am interesting in ways/suggestions on how you would start learning/hacking the new system.
Skim through the code, looking for places where you think you know what is occuring. Read through those sections to see if your initial thought holds up, and then try modifying it to see if you can change it to do something else.
I'd suggest that this be guided slightly by the project roadmap so you can then move on to spotting where new features and bug fixes are so you can contribute in a meaningful manner.
Also, read the documentation and any spin-up documents that may exist (functional specs, requirement specs, etc). This can further help you learn the code ropes quicker :)