I recently did an accompaniment version of Rainbow Connection as a lesson and this arrangement just sort of materialized. I haven't really worked it out carefully but this lesson shows a relatively easy way to play it as a solo. I encourage you to take what I put in the lesson and add your own embellishments. It is also important to work on playing each verse slightly differently, following the rhythm and phrasing of the syllables. In other words, you should be singing the song in your head and using those phrasings and accents as your guide.