1. I will learn a piano.
This makes no sense. You can learn 'a song' or you can learn 'a piece of music', for example, but you can't learn 'a piano'.
'A piano' is the physical object. You would say 'There is a piano in the lounge', for example. You can't 'learn' an object.
2. I will learn the piano.
3. I will learn piano.
These are both fine. Both of these refer to the skill of playing the instrument.