My advice would be 'it depends'. If your main goal is to learn how to use object-oriented languages, spend as little time as you can in the C-book and then move on to C++, or begin with C++ straight away. The problem is that both languages force a certain, but different way of problem solving on you. Either language supports the method of the other, but the result is never pretty. Writing OO-programs in C involves a heck of a lot of pointer wizardry; writing procedural programs in C++ inevitably abuses the OO-ness of the language, making it very hard to properly OO-ify the program later on.
While learning C++, you are faced with the aditional problem that almost no book teaches you how to properly design an OO-program. The syntaxis of C++ is not very difficult, but understanding the semantics requires a surprising amount of effort, much more than just working through the book on C++ itself. To use C++ effectively without shooting yourself in the foot (not with a big gun, at least) you require specialised literature. Despite its not-so-young age I recommend Cay Horstmann's Practical Object-Oriented Development in C++ and Java (Wiley, ISBN 0-471-14767-2) for the finer points. Read that along with your Teach Yourself C++, and you'll become a proficient OO-coder.
However, I am of the opinion that every C and C++ programmer should have torn out their hair at least once over a frustrating and exceedingly subtle pointer problem. You won't really start to appreciate languages which don't have pointers until you have had your nose rubbed in it.
