"Hot Cross Buns" is a familiar tune that you can play using only the thumb, index finger and middle finger of your right hand. You can play the song without any real knowledge of music notation, as long as you can find middle C on the keyboard. "Hot Cross Buns" features a good variety of note durations, including half notes, whole notes and quarter notes. The song features stepwise motion, meaning that you'll never have to play a note that isn't next to the key you're currently playing.
"Mary Had a Little Lamb" is an excellent second song to learn on the piano. The melody is extremely recognizable, and the duration of the tune is eight measures. Similar to "Hot Cross Buns," this song features only stepwise motion. The tune uses the same three keys as "Hot Cross Buns," and the rhythm is easy to understand because most of the notes are quarter notes.
"Ode to Joy" is another great song. Like the two previous numbers, this one features stepwise motion and a very familiar melody. This song will incorporate your ring and pinky fingers as well. It also introduces the concept of starting on a note other than the root note of the scale in which the song is written. In "Ode to Joy," you start with your middle finger and end with your thumb.
Once you're ready for a little more rhythmic complexity, try "Row, Row, Row Your Boat." It is in 3/4 time, unlike the previous three songs that were in 4/4 time. This song skips a few bar lines here and there, breaking the pattern of stepwise motion. It also requires you to cross your left hand over your right, a fun trick you can use to play the high note along with the "merrily" line.
If you're in the Christmas spirit, "Jolly Old Saint Nicholas" is a great tune to play that incorporates your left hand as well as your right. Don't worry -- you won't have to use both hands simultaneously. The song features a simple rhythm including quarter notes, half notes, and a couple of whole notes.