Here are the fingerings that I use to play a chord-melody arrangement of Bart Howard's 1954 standard Fly Me To The Moon. I play it in the key of G not the usual key of C. The song is played in a medium 4/4 time. Alternate chord substitutions are indicated in parentheses above the chord I used.
This song is a great example of a Diatonic Circle Progresion. Diatonic Circle Progressions are a form of Circle Progression that is constructed using only those chords from the diatonic scale.
Dominic Pedler in his book "The Songwriting Secrets of The Beatles" talks about the diatonic circle of fifths like this: "If there is one winning formula at the 'poppy' end of popular music, it is those progressions whose roots follow a predetermined movement descending in intervals of a fifth... Applicable to jazz and classical music as well as to pop, the cycle is an unashamedly formulaic system for exploiting the natural tendency of diatonic chords to follow this predetermined path. It provides a very simple way of appreciating how generations of classic pop songs actually work in practice..."
Circle Progressions allow numerous tritone substitutions that can add interest to your chord-melody arrangements. This approach was used in the second "A" section below.