IMHO, beginning with getting a firm grasp of the basics of C will help you in the long run when delving into C++. As for where to begin learning C++, I think that it's best to go for just a general knowledge of C++, then going to a specific area of programming style, such as game programming.
If you want a mixture of good knowledge of C++ as well as the basics for game programming, I'd recommend the book 'C++ Programming for the Absolute Beginner' by Mark Lee. It teaches you how C++ works just like a normal book would, but using game analogies and examples to teach you, so it's like hitting two birds with one stone.
However, it still stands that you should learn some C before C++. I started off messing around with C, in which I developed my first text-based game (which I got rid of last year). After that, I read the aforementioned book, and used the net to learn C++, and I'm in the process of making another text-based (soon to be 2D grapical! ) game.
But still, you should just do what suits you best. All we can offer are pointers and advice.