I don't mind admitting that, while I was a graduate student, doing the Legendre transform in statistical mechanics was more of a "automatic" response rather than anything I actually understood. All I knew was that it got me from one place to the other, and that's that. Luckily, I don't quite use that piece of information, and that skill, in my everyday work. Unluckily, it means that, while I can still tackle such problems, I don't think I have that good of a grasp of it as I do with other aspects of physics.
That's why I was rather interested in reading this preprint on an effort to clearly introduce the Legendre transform. The authors used the standard, useful examples from classical mechanics and statistical mechanics, so physics majors should be well-familiar with the coverage. I've only looked through it rather quickly, with the intention of reading it more carefully later, but I think this could potentially be highly useful to many, especially if you are still in school learning this subject.
If you have read through this more carefully, and have some comments, please post them here. I'd like to hear them.