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I'm going to use this song as more of an exercise to train your ears than a song lesson. To start off, let's just pick a note. We'll use D. Sing the song and pay attention to the pitch of the notes. We know the first two notes are the same, and they make up the first twinkle. The notes change however for the second twinkle. They go up in pitch a little.
Well, let's start making our way up the scale and see which note fits. You'll work your way up the scale until you get to the fifth interval, A, before you find the right note. So we know that the next twinkle in the first verse is made up of two A notes. What about the three notes that make up "little star"? Well, if we sing it we'll find that the melody goes up a little for the word little and down a bit for the word star. So, for the first verse we have D D A A B B A. The best part is, we didn't need sheet music to tell us that. We figured it out using our ear and our piano!
The melody comes down in steps. You can try playing it as G G F F E E D, but that won't quite sound right. That is a good hint that the song is not in the key of C. Listen closely and see which note sounds off. It's the F. Try playing the same phrase with an F# instead of an F. It sounds more correct doesn't it? We also notice that our ear is sort of pushed to the D when the melody is playing, so there is a good chance that our song is in the key of D major. Knowing that the song is in D major let's us know that we will be using the C# and the F# instead of the C and F.
Finished off the rest of the song by listening to the pitch as we sing and picking out the notes on the piano gives us D D A A F# F# D as the notes for "like a diamond in the sky".
Now let's add some chords to the song. We know that the root is D. So start the song with a D major triad. When the melody gets to the B, we need to change the chord, but which one do we use? We want a chord in the G major scale that has a B in it. The E minor chord has a B. So do the B minor chord, and the G. Let's try all of them and see which one fits better. The E minor chord doesn't really sound right, neither does B minor. The G chord, however, fits nicely. Let's change back to a D chord when the melody hits the word 'star'.
Starting the second verse is a G. So let's start that verse with a G chord and move back down to a D when we get to the F# in the melody. When we get to the E, we have a number of choices. E minor contains and E note. But so does A, and A is the fifth chord of the scale. Five chords really like to lead back to the root, which is what we want. So let's use the A here. Now repeat the song from the beginning.
The key to this exercise is to use your ears to hear which chord is best in which spot. We are using simple triads now, but there are many other chords which add notes on top and can give more nuance to your playing. I'll teach you about some of those in later lessons.
For the next song, you can practice London Bridges Falling Down.