From the arduino.cc home page:
The microcontroller on the board is programmed using the Arduino programming language (based on Wiring) and the Arduino development environment (based on Processing).
Now, we all know that description by "Team Arduino" is a bit of stretch -- the "Arduno Language", if it exists, is a certain amount of IDE preprocessing of the .pde/.ino "sketch" files that convert certain non-C++ Arduino specific stuff into "real" C++ that the g++ compiler can actually compile.
But it should be made clear that when you program in the Arduino IDE, there are enough differences that mean the syntax and semantics depart from standard C++ (in particular, the the C/C++ preprocessor), and these can lead to unexpected (and sometimes inconsistent) results that can be confusing to anyone who has actually programmed in C/C++ before.
Perhaps calling the "Arduino Language" an attempt at a simplified "dialect" of C/C++ is a fairer description?
I just think of the differences as simply "broken", mostly. But the "team" will say that substituting the standard keyword "bool" with "boolean", and saving the user from having to declare their own function prototypes, and hiding compiler warnings and error messages from users so as to not upset them, are helpful to novice programmers. So all a matter of taste, I suppose. I myself actually program Arduinos without using the IDE, so I can stay in standard C++, and avoid all the non-standard Arduinoisms.
As to the question of learning C or C++ first, I'd recommend learning C first. It's a smaller, simpler and cleaner language, and in many ways more coherent than C++, which is (relatively) vast, complex and untidy. Once you've mastered C, it's a matter of learning C++ incrementally as a series of extensions the basic core of the C language. Almost all of C is necessary to program effectively, much of C++ is not.
In my opinion, the first and last book you'll need for learning C is "The C Programming Language" by Kernighan and Ritchie. Like the C language itself, the text is small, simple and clean. A classic.