My right hand has been hurting a lot recently after taking notes and playing piano. Should I be worried?

Not really. If it is sore after excessive activity, then it is probably sore tendons and muscles in the hand. Should get better with a day or two of rest.
Hand pain. You may be suffering from repeatative motion condition causing inflammation of the tendons and soft tissue of your hand or a nerve entrapment at the wrist (carpel tunnel syndrome). You should consult your doctor concerning diagnosis and treatment. Meanwhile decrease the irritation of repetative motion as much as you can, especially the note taking, and be sure your posture is correct at piano.