I learned C first, and I took a course in data structures which used C, before I learned C++. This has worked well for me. A data structures course in C gave me a solid understanding of pointers and memory management. It also made obvious the benefits of the object oriented paradigm, once I had learned what it was.
On the flip side, by learning C first, I have developed some habits that initially caused me to write bad C++ code, such as excessive use of pointers (when C++ references would do) and the preprocessor.
C++ is really a very complex language with lots of features. It is not really a superset of C, though. Rather there is a subset of C++ consisting of the basic procedural programming constructs (loops, ifs, and functions), which is very similar to C. In your case, I would start with that, and then work my way up to more advanced concepts like classes and templates.
The most important thing, IMHO, is to be exposed to different programming paradigms, like procedural, object-oriented, functional, and logical, early on, before your brain freezes into one way of looking at the world. Incidentally, I would also strongly recommend that you learn a functional programming language, like Scheme. It would really expand your horizons.