How important singing is for piano playing
I can't address enough how important singing is for piano (or any other musical instruments). Some of my students (especially beginners & intermediate students) don't want to sing even if I require them to sing while playing some passage. They say, "Oh, I'm shy.," "No, I don't have to sing. I'm OK without singing.," or "I'm a bad singer." Well, it doesn't matter what you think about your singing. Even if you are a terrible singer, even if you can't sing in pitch, I encourage all piano learners to sing what they are...
Use your entire body to play the piano.
A lot of beginning (or even intermediate) piano players play the piano with only fingers like typing on the computer keyboard. A piano is a huge instrument. It has 88 keys. It's long. To tackle this big instrument, you need to play with your entire body, just like sports. It's important to support your body with your two feet. Sit on the edge of the piano bench so that you can lean towards the piano, not sitting straight parallel to the piano. Come on! You are dealing with this instrument....