Just do it. Switch from one key signature to another in mid song. Insert a slow section. Change to another time signature. Your changes may not be well timed at first, but you'll never get the hang of it without practice. Just start by making changes at arbitrary times.
Take a look at David's lesson, 'Ch-Ch-Ch-Ch-Changes'
Switching key signatures will produce a key change. I recommend that you just make a switch. Move to a key signature that's a half step or a whole step higher. E to F, or A to B, for example. It won't be graceful, but what's important is that you will have done it, and it won't be quite so mysterious. In the next attempt, worry about musicality a little more. Use a pivot chord, like the article says, and go to the relative minor (no key signature change in that case).
Switching key signatures will produce a key change. I recommend that you just make a switch. Move to a key signature that's a half step or a whole step higher. E to F, or A to B, for example. It won't be graceful, but what's important is that you will have done it, and it won't be quite so mysterious. In the next attempt, worry about musicality a little more. Use a pivot chord, like the article says, and go to the relative minor (no key signature change in that case).
I'm not familiar with the song. Keep in mind that he had a lot of practice writing songs, and he didn't switch keys four times in a song without having written a lot of songs with fewer changes. Don't try to start at a high level. Keep it simple until you get the hang of it. Without practice, it is hard.
Try looking for common notes or chords that you can use to bridge sections.
For example, both the keys C and F major have an F major chord. This could help make your transitions smoother.
Take a look at an orchestral score for a symphony sometime.
You'll see 20+ instruments going through a half dozen key and tempo changes in some cases.
It's not that hard, just do it.
Read the article, it does a nice job of outlining what the basic ideas are.
Start with tempo changes, those are easier. Write a 4/4 song with a 5/4 chorus
Now once you like it, take that 5/4 chorus and shift it to the relative major or minor of the key of the rest of the song.
Now add a bridge to the song, it doesn't have to be difficult. Shift it up to the 5th of the chorus. Maybe change teh tempo to something like 8/8, or 6/4.
Now, take a brand new song, and do it again.
And again.
And again.
Once you have the basic idea, start going for more complex changes using bridging chords
"The music business is a cruel and shallow money trench, a long plastic hallway where thieves and pimps run free, and good men die like dogs. There's also a negative side." -- HST