I'd like to show students a graph of a function, say $(x,y)\mapsto\sqrt[3]{x^3+y^3}$, and then highlight its various properties connected with (non)differentiability at certain points etc. Having some lines included in the graph seems a good idea. Now I'd like to be able to highlight (by using different colors and/or thicker lines) some of these lines. It would be best if I could press or click something so that one or another line gets "selected" (highlighted). How to achieve something like this?
Note: I'll be fully satisfied with one-line, RTFM-style answers, if someone could provide me a hint where to look for the FM; I'm new to Mathematica and feel a bit overwhelmed by the multitude of documentation in various formats...