Learn the lyrics. The lyrics of "Zip-a-Dee-Doo-Dah" are simple and fairly repetitive. The complete tune is:
Zip-a-dee-doo-dah, zip-a-dee-ay
My, oh my, what a wonderful day
Plenty of sunshine coming my way
Zip-a-dee-doo-dah, zip-a-dee-ay.
Mr. Bluebird's on my shoulder.
It's the truth, it's actual,
Everything is satisfactual.
Zip-a-dee-doo-dah, zip-a-dee-ay
Wonderful feeling, wonderful day.
Learn the intervals. The first part to learning how the lyrics fit into the song is to know which pitches they hit, as well as where one note goes when it moves to another note. The intervals will either move to a higher note or lower note, each which is defined by the next pitch it goes to. For instance, the word 'zip' to the word 'a' goes up one interval step. You can start on any note you want, then go up one pitch to the next note.
Listen to the melody. Once you have an idea of whether your pitches are going up or down from word to word, find a space that offers the complete melody to the song. You can follow along with this and sing the words as well as the intervals. This way, you can tie in the intervals and the rhythm that is a part of the melody.
Learn the rhythms without the melody. While you are listening to the melody, don't just incorporate intervals but add rhythm. Each of the words will be placed with a beat that is in the song. Making sure you line up the direct rhythm ensures you get the right sound when you are singing the song. You can easily find your rhythm by tapping your foot to the song. Whenever you hear an accented word or beat, that's the main beat of the rhythm.
Put it together. From listening to the rhythms and melody, and figuring out how it fits together with lyrics, you can begin to sing the song on your own. The combination of these provides you with a simple and effective way to sing one of the happiest songs written.