I'm no great c or c++ programmer - but I really like Bjarne Stroustrup's "Learning Standard C++ as a New Language" (C/C++ Users Journal. pp 43-54. May 1999):
http://www.stroustrup.com/new_learning.pdf
Even if it is somewhat dated now - I think it makes a great case for c++ (or d, maybe rust) over C - and it doesn't even touch on Unicode - simply memory management, error checking and buffered I/O.
I agree everyone should know some C - and some assembler (if for nothing other than looking at compiler output) - but I'm not sure I agree that C is "simple". And learning from k&r might not be the best approach today - rather read sources of something like some of the openbsd utilities, redis, spiped...
C was probably the first language I learned, but I wouldn't say I "know" it. I still have no idea which calls are safe or preferred or how to structure a larger program.
It depends what you want to get out of it. If you want to learn for its historical value, to learn more about the language that a majority of other languages adopted ideas from, K&R is probably good enough. If you want to grok a bit of random C code, that might be good enough, too, but you'll probably need a bit more. If you want to contribute to the Linux Kernel or similar project, or even put C on your resume, that's definitely not enough.
You could make an even stronger claim about knowing an assembly language. The point being that understanding how a computing device works is helpful in getting the best out of it. If that wasn't your point, then C is no more special than any other popular programming language.
Also, whilst I only know the basics of C, it's pretty clear things have moved on considerably since the days of K&R. It doesn't cover any of the enhancements included in C99 and C11, nor does it cover the tooling and coding conventions (the ones that help avoid problems) that have become commonplace since 1989.
For a more up-to-date reference, I've heard good things about 21st Century C:
It wasn't; it is perfectly possible and normal to use C as one would any other programming language, i.e. without knowing much about the computer architecture. Even the pointers, often seen as something low-level, aren't really: the rules of their behavior and use are very simple, and they are easy to describe at a high level.
C is special in that, as I said, it has been, and remains, a very important programming language in its ubiquity and sheer power.
In that case, you could say the same about Java, Python, C#, etc... If being a "true" programmer is just about getting the job done, any of those languages will suffice. C may be a useful tool for certain performance critical code, but that's a very small fraction of what's required in most day-to-day programming activities. I would say the following list describes the key qualities of good code (in the order described, from most to least important, for most cases):
Correctness, clarity, conciseness, performance.
As for pointers, you can describe them at a high-level but to understand why they're a low-cost abstraction it's helpful to understand how computers work. That's the type of knowledge that C can help expose you to.
After working through the problems, two books that would be great followups are Hanson's C Interfaces and Implementations and Bryant and O'Hallaron's Computer Systems: A Programmer's Perspective.
Nevertheless, I don't think anyone should call themselves a (true) programmer unless they know C.
C is easy to learn, because it's a small language. If you are not new to programming, pretty much all there is to know is in the second edition of the K&R book.