I think that learning C before C++ might hurt you.
C teaches you something very useful, which is how the computer works under the hood. Most things you do in C loosely match to what the computer does, and learning C helps you understanding the computer.
I'm not saying that it's a good language to start with, but it's okay. Unlike C++.
IMO, C++ is the worst possible language to start with. The reason for this is that C++ tries to support everything, often using handy but cryptic syntax. C++ doesn't guide you to program one way or another, it simply adapts to every possible way to solve the problem. This might be good on the hands of someone that already knows how to solve the problem, but it's completly overwhelming for a begginer. It feels like you're taking the wrong turn at every step.
A much better alternative would be JavaScript. Yes, it does have flaws, but it's much simpler and has personality.
I wasn't judging C++ as a beginner language. I wasn't judging C either.
IMO the problem isn't so much about learning C or not. It's about which other languages do you learn alongside it.
You need to practice in a language that encourages OOP and a little bit of functional programming, like JS or Ruby.
If you know Ruby and C already, using C++ well will be much easier :)
As long as that's remembered, I think the opposite is probably true: knowing C helps a lot to learn C++ (and vice versa) as opposed to learning either from scratch.
True, modern C++ is quite different. But, there's nothing from preventing a C programmer from starting as a "bad" C++ programmer and learning new techniques as they go.