Most three year olds are not ready to start learning piano in the conventional way. They generally do not have the focus or dexterity to learn to read and play music on a piano. That being said, there are ways you can begin teaching your three year old to play piano, especially if he is demonstrating an aptitude and interest in doing so.
Encourage your child to play the piano for fun. Give her time and freedom to play however and whatever she wants. Do not force your child to play if she is not interested. This will cause her to resent having to play the piano and can have a very negative affect. If your child is asking questions about playing the piano, this is a good indication that she is ready to learn more. If your child is just having fun, plunking on the keys, let her enjoy her musical experience on her own terms.
Model playing the piano for your child. If you do not play the piano, then learn a simple song and play it. Also, take your child to see other people play the piano, but do not force him to sit for long recitals or concerts.
Tape letters and pictures on the piano keys. Put the letter A on the A keys and a picture of an apple. Put the letter B on the B keys and a picture of a bear. Put the letter C on the C keys and a picture of a cat, and so on through to the letter G. This will help your child start to associate the notes seen on the piano with the keys' letter names. Take time to point out the letters, say the name of the key and play the notes with your child.
Have your child practice rhythm on the piano with some basic piano exercises. Play the rhythm for "strawberry, strawberry" or "shave and a haircut, two bits" on the same note. Have your child do it after you. Then do the same thing on another note and another and so on. This activity also strengthens children's fingers.
Sing a note and see if your child can find that note on the piano.
Teach your child a simple song like "Hot Cross Buns". If your child does not seem interested or is getting very frustrated, do not force the issue.
Get your child involved in a Kindermusic program. This will help her develop a general love for music and understand the basics of rhythm, tempo and movement to music.
Purchase a program that teaches preschoolers to play the piano (see Resources). These piano-playing methods may require you to tape different symbols on your piano keys.