Normally John Denver played this with a capo at the 2nd Fret. We've kept it in open position for ease. You can still use a capo if you wish, the fingerings all remain the same, just up by 2 frets.
The song follows a loose style of country fingerpicking. That means that there is not a regimented pattern (like you would find in some classical music). It is loosely based around the T1T2 fingerpicking pattern. That is the most prevalent pattern, and very common in country styles. That's partly because it emphasises alternating bassnotes (played by the thumb), which is a staple of the country style. However there is quite a lot of deviation from the pattern throughout the song. It doesn't start with it straightaway for example.
This loose T1T2 pattern is applied to a pleasant chord progression. The chord sequence is repeated a lot, so don't worry too much about this being a very long song to learn. Once you know the main chord sequence, the rest is the same (albeit with the odd bit of variation). In fact the intro is the most varied part of the song. For that reason we recommend learning the chord progression from the first verse, just after the intro. Once you know that, the intro will be much more straightforward. Besides, if you know the main progression, you can choose to learn the intro or skip it.
Main Chord Progression (After the intro):
C major - G major - A minor - E minor - F major (add D + G open notes) - G add4 / G major
These same chords are played again, but differently the second time:
C major - G major - A m7 (then add G in the bass) - F major (add D + G open notes) - G add4 / G major (repeat this 2nd sequence a few times).
That's the whole song in a nutshell. Once you know those 2 parts the whole thing is just more of the same. Even the intro (which as we said earlier, is the most varied part of the tune).
If you are new to fingerpicking, or find it hard with these swift chord changes, make sure you work through the Core Skills section on Fingerpicking. The T1T2 country picking style is included in there, and a whole lot of useful exercises designed to help your fingers get used to it!