Right now I'm getting familiar with C and the C standard library and I wonder if my knowledge in this area will be useful when I turn to working with C++ at a later time.
Therefore I'd like to know, whether I can use the functions provided by the C standard library in a C++ setting, and also whether and why it would make sense to actually do so.
Yes, C++ was originally designed so that any C library can be easily used in C++. Of course this is slightly less true (in particular, if a C library happens to use some C++ keyword like try or dynamic_cast, it won't work; also, if a callback coded in C++ passed to a C library is raising some exception, you are likely to have a big mess).
The standard practice to use a C header file in C++ is
extern "C" {
#include <some_c_header_file.h>
};
and most existing C header files are designed to cooperate with C++ by actually containing stuff like
#ifdef __cplusplus
extern "C" {
#endif
//// most of the header material goes here, C style
#ifdef __cplusplus
}; // end extern "C"
#endif
In practice, many C standard headers have equivalent C++ headers wrapping things like above (and also in namespace std). Eg C <stdio.h> is C++ <cstdio> -but you often should prefer genuine C++ streams (<iostream>), however printf-like routines are usually more localization friendly mixed with gettext(3).
However C and C++ are very different languages. You should code in idiomatic C++11 (using standard C++ containers, auto, closures, RAII, smart pointers, rule of five, SFINAE, exceptions, anonymous functions, ...)
Some standard C functions are not very useful in idiomatic C++. For example, you are unlikely to use directly malloc in genuine C++ (at least prefer new -which is still very low level and no more in the C++ spirit-, more likely use a lot the containers and the smart pointers without dealing manually with heap allocation). But POSIX functions (notably syscalls(2) ....) are quite useful in C++. longjmp is likely to be incompatible with C++ exceptions.
BTW, C++ has evolved a lot in this century. Don't learn C++98 but at least C++11 (there are tremendous differences between them) and perhaps C++14. Use a recent compiler (GCC or Clang/LLVM); in december 2015, that means GCC 5 at least or Clang/LLVM 3.7 at least. Don't forget to enable all warnings & debug info in the compiler (e.g. g++ -Wall -Wextra -g -std=c++11)
C++ (that means C++11 at least) is a difficult programming language, considerably more complex than C is. You'll need weeks of reading to learn some of it, and good coding style and discipline is essential (you can easily write very crappy code in C++). Start with Programming: Principles & Practice Using C++
I believe that if you only know C, reading SICP (and studying a bit of Scheme) before learning C++ is worthwhile.
The notion of undefined behavior is very important, both in C and probably even more in C++. You absolutely need to understand it (see C.Lattner's blog on it) and avoid it.
You will also learn a big lot by studying (and perhaps contributing to) some existing free software and its source code. Hence I recommend using Linux.