I believe that learning C will most definitely help you be a better programmer. A majority of the software you use (including perl!) is written in C -- it can be powerful, portable (mostly), and fast. As others have said here, truely grokking C will force you to have a better understanding of how the computer works (e.g. pointers, memory allocation, etc.) It's a step up from assembly language (which I also recommend learning) but still close enough to the hardware to make you understand your machine. A quote I once heard was "C -- all the power of assembly language with the ease of use of assembly language." heh heh.
The book I most recommend for someone who already programs but wants to learn C is Kernighan and Ritchie's The C Programming Language. Kernighan and Ritchie invented C (and did a fair amount of the original Unix OS I believe).