There are indeed so many things to learn, but as a developer or someone who would be in the development world (be it building tech startups, developing software, etc) there is a must-know set of tools. C/C++ happens to be very high on the list. And learning more really doesn't hurt.

Not just C++, but also learn C, Java, Perl, Python, Erlang, Lisp, PHP, Shell Scripting, Assembly, Ruby, along with their most used libraries/frameworks. The more diversify your skillets, the better you can use the right tool for the moment, improving performance of that, lower development time. (Eg. A spoon can be used to build a house, but it's hardly the best tool out there).

This way, want to hack up a quick content management or low computational front end system, whip out your php/python/ruby with your favourite framework out. Got a bottleneck at high cpu or memory bound place, rewrite that piece in C/C++. Need to deploy a cross platform app really quickly, Java. Want to bind or write a test for a bunch of tools, use Python or Perl. The list goes on.

Summary: Learn More, use the right tool, do less, get more accomplished.

"C/C++" is not a single language, by all means learn C.

One should learn both. If you spend the time to learn C, might as well spend a tiny bit more and learn the C++ concepts as well. There are ton of pure C libraries, but a lot of algorithmic stuff in C++ STL/Boost kinds, knowing both C and C++ is essential, each by them selves wouldn't necessarily help one too much.