I was taught C at college (well, they gave me a textbook and a compiler). Sadly this was long after C++ was standardized but the lack of decent CS tutors in the UK means that the syllabus is still largely dictated by the available skills of staff rather than the real world, in smaller colleges at least.

Bizarrely, even though they were teaching us C, they had us using Borland C++ Builder, so the template code we were filling bits in on contained all sorts of gibberish that made no sense to me. The textbook they supplied had a small, very badly written appendix on C++ so a combination of that and curiosity about all this weird code Builder was generating made me start to experiment in my own time.

Thankfully I left the course before I wasted too much time learning outdated methods, and I wouldn't recommend learning C before C++. I spent a long time implementing my own string and vector classes which was a great way to learn about things like operator overloading and RAII, but produced terrible code. I'm fortunate that I was able, largely through activity on this site, to throw off most of the bad habits I got into in the early days.

Before I had even looked at C though, I had spent most of my life (from about aged 7 onwards) playing around with various BASICs. ZX Spectrum, followed by some game BASIC for the Amiga. It was very clear to me going into the course that having this fundamental understanding of things like variables, control flow, input etc gave me a big headstart on the students who had never programmed before.

I think we have to draw a distinction between learning programming, where the skills should allow you to move from one language to another easily, and learning the nuances of C++. Personally I think my basis in C made it easier to understand some of the peculiarities of C++. I'd already started playing around with writing simple compilers and virtual machines before I got into C++ and I find a bit of understanding of how compilers work under the hood helps make C++ make a bit more sense, but that isn't a requirement.

I learnt C++ after coding in C for years at work to modify a COM/ActiveX control. I practiced a bit and read Bjarne's book.