One of my first professional programming projects was to take a look at the custom C++ billing software our company had purchased from a contract programmer.
I had a long unix and programming background, and was back for a summer job after doing 1 semester of C++ in college.
My boss told me, since I was the only one who had C++ experience, to start documenting the system.
At the time, we were using IRIX, and so I was using the SGI compiler and tools suite, which were, I believe, licensed from EDG. The point is that there was a very nice call graph visualizer. This was helpful for understanding things at a superficial level.
However, what was even better was just running the program a bunch of times on test data and seeing what it did while under the debugger.
While my summer began with the task of documenting the system, as I learned things I'd report them to my boss.
By the end of the summer, I had re-written some fundamental parts of the system; I'd moved some of the processing outside, and I pre-processed and pre-sorted the data.
The overall execution time went from many hours to about 45 minutes to calculate monthly bills. THe key innovation was replacing the inner loop of the charge tabulation -- which was 2 or 3 levels of nested linked list traversal.
Instead, I used the standard unix sort tools to pre-sort the data files before being loaded into the system, and I changed the system to use a data structure that supported a binary search.
The majority of the code got left alone. By understanding the code under a debugger, and realizing that how it worked on production data was much different than how it performed on the test data it was originally delivered with, I was able to make a critical set of changes that had a huge impact.
In general, I spend as much time as I can not writing code, but instead, understanding how the existing system works. For a current project, I've spent the last two weeks playing with somebody else's code, and now I've expanded it so that it can also operate on my data sets, and I've probably changed fewer than 100 lines across about 5 different projects.