I’m a programmer with 2 years experience, I worked in 4 places and I really think of myself as a confident, and fluent developer.
Most of my colleagues have CS degrees, and I don’t really feel any difference! However, to keep up my mind on the same stream with these guys, I studied C (read beginning C from novice to professional), DataStructures with C, and also OOP with C++.
I have a reasonable understanding of pointers, memory management, and I also attended a scholarship which C, DataStructures, and C++ were a part of it.
I want to note that my familiarity with C and C++ does not exceed reading some pages, and executing some demos; I haven’t worked on any project using C or C++.
Lately a friend of mine advised me to learn C, and C++ extensively, and then move to OpenGL and learn about graphics programming. He said that the insights I may gain by learning these topics will really help me throughout my entire life as a programmer.
PS: I work as a full-time developer mostly working on ASP.NET applications using C#.
Recommendations?
For practical advancement:
From a practical sense, pick a language that suites the domain you want to work in.
There is no need to learn C nor C++ for most programming spaces. You can be a perfectly competent programmer without writing a line of code in those languages.
If however you are not happy working in the exact field you are in now, you can learn C or C++ so that you may find a lower level programming job.
Helping you be a better programmer:
You can learn a lot from learning multiple languages though. So it is always good to broaden your horizons that way.
If you want more experience in another language, and have not tried it yet, I would recommend to learn a functional programming language such as Scheme, Lisp, or Haskell.