Many users download and install software from 'tarballs'. These files are archived directories of source code. The extension is usually tar.gz or .tgz. Any project in the sourceforge website will also have source code available. Also, you could install a source package. These are rpm packages that contain the sourcecode. Look for files the end in .src.rpm. This is the standard suffix for source RPM packages.
If you visit the ftp site where you can download your linux distribution, there will be a directory that has nothing but source code. Because the GPL license requires one to release the source code, this is what open source is all about, you won't have any problem locating source code for linux.
One source package you might want to look at is the core-utils package. This contains many ubiquitous programs such as 'echo' and 'cat'. The Unix V7 echo program is only 23 lines long, including the blank lines.
You didn't ask for this, but I would highly recommend the book 'Linux Programming by Example: The Fundamentals' by Arnold Robbins