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How do you chord progress go from the key of C major to the key of A minor?

I would do it this way: progress to the III chord (Em), change the III chord to a Dominant 7th form (E7), now the E7 chord will will want to resolve to the I chord in the new key (but is the new key A or is it Am??).

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There's lots of ways to do it! No single way is necessarily the right way. Your progression certainly works, and you're right that preceding the Am chord with its dominant (either an E or an E7) is the best way to strengthen the effect of Am being the new home key. You can actually follow the E (or E7) chord with either an A minor (A, C, E) or an A major (A, C#, E) depending on which key you want to go to.

Rhythmically, another thing that can help to strengthen the effect of the modulation is to place the resolution to A minor (or A major) at beginning of the following phrase. So, for example, if your phrases are four bars long (with one chord to a bar), then end the fourth bar with the E chord, and then the next phrase can start in the new key of A minor (or A major if you'd rather). Then your progression might look like:

C _ _ E7 Am ...

where the two "_"s can be filled in by pretty much anything. I've tried various combinations of F, G, C, Am and Dm, and they all seem to work pretty well. In fact, I haven't found a combination yet that doesn't work. Play around with it.

I would consider preceding the E7 with something other than an Em, though, just for the sake of variety. Unless it's the specific sound you want, pretty much any other chord will add something more to the progression.

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You've got the way to do it. And the new key will be A Minor (unless you want to go to A Major!). E7 acts as a dominant 7 (V7) chord in both A Major and A Minor, so will effectively establish either key with a perfect (V-I) cadence, or V7-I to be precise.

Although the key signature of A Minor doesn't have a G# in (it has no sharps or flats, being the relative minor of C Major), the Harmonic Minor is used instead of the Natural Minor precisely to create a major dominant chord, which strongly establishes the key with a V-I cadence. The G# in A Harmonic Minor can be thought of as both the leading note and the Major 3rd of the dominant chord.

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The way you, Bob and Caleb would use is fine, and probably the most used way of getting from a major to its relative minor. Absolutely a great way, found in many tunes. But try this for something a little different. The bar before Am, use a diminished chord. B is one that works well. Or D or F or G#. They are, of course, one and the same, depending which note is at the bottom. It works well as the B moves a semitone to C, F moves a semitone to E, and G# moves a semitone to A. All tiny changes which make the following Am almost inevitable !

That diminished chord could be thought of as an E7b9 -another option- without the E.

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