Lots of great advice here! I would add two things:
Never play an instrumental piece faster than you can play the hardest passage with accuracy. KEEP IT SLOW, and practice slow, over and over. Your speed will pick up naturally as you become more comfortable with it.
After you've learned a piece, make yourself play or sing it with your eyes closed. When we remove the visual, all that is left is the aural, and you can REALLY focus on what you hear.

Robin