Well, luckily, you don't have to solo over the intro to Proud Mary... so I've never really analyzed it that closely.
To move a major chord by a 3rd (either major or minor 3rd) is called a chromatic mediant move (The "mediant" is the fancy name for the iii chord, FYI.)
It comes up a lot in blues when you think about stuff like La Grange going from A to C to D, which comes from John Lee Hooker doing it from E to G to A. In "China Grove" by the Doobie Brothers, the last part of the verse right before the chorus goes from A to C also - it's fairly common in modern music.
So the C A C A C A at the top of Proud Mary is called a chromatic mediant, that's easy to name. The A, G, F part is what I call the "Lola" progression because Lola is the first time I saw it (C D E at the beginning of the song.)
However, we could also bring in some jazz theory - the rule is that any minor chord can be made into a 7th chord (and a major chord is just 3/4 of a dominant 7th chord) so you could say it's a I VI I VI I VI V IV in C - and that's probably how I'd describe it to someone if I had to.
From the last F it jumps to D and then clearly moves to that key for a while. F to D is, again, a chromatic mediant relationship so that's not hard to accept.
Once on the D, the verse is D for a while, A, Bmin which is all in the key of D easily. Fortunately, if you solo over this tune that's what you'll solo over and it's straightforward. In the Tina Turner version (which we play in Yard Sale) it's D, then A, Bmin, G, and I usually play the G as a G7 which distorts the key center even more but I like it and I think I hear it in there somewhere so I do it.
That's about all I got on that one - it's a pretty oddball chord progression... almost more of a melody or a riff than a chord progression.