EDIT: I never MEANT to get involved with this thread, I just took issue with the idea that C was akin to COBOL these days... that's just not the case. The 'chart of commits' from that random project hosting site that was cited showed almost no C commits, as if it were 'dead'. So, that's what I took issue with. It's far from dead, as you can see. Heck, *whatever you are reading this thread with* is written in C or C++. The browser and OS, no matter what platform. Now, if iOS, it *might* be Objective C, but for now let's just call C/C++/Objective C the same 'family'.
As I've said repeatedly, the most important thing is that you select the language that works best for your task, and for YOU. For instance, since I don't do much C#, I could probably create a native C++ GUI faster than a C# GUI, even though it is much easier to create GUI in C# with Visual Studio. So, for me, C++ is a great thing to use, even if that means I sometimes must work a little harder. It also does continue to offer to most superior performance to any other language, if that is an issue for YOUR application. Likewise, it is USELESS for Web 2.0 stuff, so obviously not even a choice there.
MANY languages have their syntax derived from C, and so by learning C, you not only learn C, but also can easily pick up lots of other languages. PHP, for instance, has a C like syntax... though also supports other syntaxes, which makes it super easy to pick up on, one of its advantages.
I also recommend all students study assembly language (x86, ARM, doesn't matter).. any assembly language, just so you understand how native code gets compiled and executed by the processor. Then, you can understand disassemblies. Assembly language, in contrast to its reputation, is actually more SIMPLE than any other language. That simplicity means it is harder to actually do things WITH though.