"How High the Moon" is one of those standards that has been covered by so many artists, it is tempting to think it is an easy song. Indeed, the melody is very straightforward. And the song form is a very conventional 32-bar ABAB. And there aren't really any strange chord changes.
But don't be deceived. While the changes seem very natural, this is a slippery song. In particular, it is loaded with cases where a major7 chord is followed immediately by a minor7 chord on the same root note. That major-minor exchange sequence comes up somewhat regularly in the American Songbook. When you see that, you can be almost certain that the key center down one whole step.
Why is that? Let's take a look at the first 5 measures (in concert key):
Gmaj7 Gmaj7 Gm7 C7 Fmaj7
The first two measures are simply the main key center of the song: G major. In bar 3, we go from G major to G minor. That Gm7 begins a ii-V-I sequence in the key of F. So as soon as we hit that that minor7 chord, we are immediately in the key center one whole step down. And that major-minor exchange continues throughout the song -- each time lowering the key center a whole step. We see the exchange happen on bars 3, 7, 18, and 22. Each of those ii-V-I patterns is 4 bars, so exactly half the song is part of the major-minor exchange pattern. The key point is to hit the new key center on the ii chord (bars 3, 7, 18 and 22), not a couple of bars later.
There is another complication that makes this song tricky. The A sections are identical. The B sections are almost identical, with the big difference being the minor chord (Gm7) in bar 11 versus a major chord (G maj7) at the same point in the second B section at bar 27. What that being the only real difference from the first AB to the second AB, you must clearly hit the minor the first time and major the second time, as that really defines the whole song.
One final point of trivia with this song. It has essentially the same changes as Charley Parker's Ornithology, although the melodies are radically different. This arrangement superimposes the Ornithology line on the last chorus, just for fun.