The language supported by the Arduino IDE is basically C++ with some additional features implemented by the Arduino programmers. Also, in a sketch you just code the setup and loop routines (there are a few others that you will eventually get to as you become a more advanced programmer).
In a sketch you can define classes in a library and include that library using the Arduino IDE. The Arduino IDE implements an Atmel compiler that creates code for the Arduino's processor (there are several models). You can work outside of the Arduino IDE (sounds like you are) but you still need to have a compiler that targets the correct Atmel processor.
Finally, C++ classes can become large, so at some point your source will outgrow what the processor can store. So, the Arduino classes are sparse and to the point!
To start with, I would use the Arduino IDE and write sketches (which are mostly C++ anyway). And as the occasion permits you can code your own libraries in C and/or C++.
