In addition to the fine answers already posted, I'll add this: the two ways of thinking about it (modes vs. chords) are exactly the same.
Imagine the chord progression is
CMaj7. Someone tells you, "solo over the
Dmin chord using the 3-5-13." What numbers is this person referring to? The notes of the
D dorian minor scale.
Or imagine that you're transcribing a solo. You take a short lick that occurs over the
Dmin7 chord and you want to learn to play it in all 12 keys. So you write out the lick out as numbers: 2-3-4-5-7-5-6-4-3-1. What do the numbers refer to? The degrees of the dorian minor scale. (
We can think of chords as being built from underlying scales. Modes are the scales from which chords like
G7, and many more are built. If not for modes, we would have to talk about the tones of a
Dmin chord by referencing the notes of the
C Major parent scale: the note
D would simultaneously be (a) the root of the
Dmin chord and (b) the 2nd scale degree of the
C Major scale. This would obviously be untenable. Having scales that start on the same note as the chord's root unifies our theory so that (a) the scale degree and (b) the chord degree both refer to the same number.
So, the two approaches are equivalent because the numbers refer to the exact same thing.
When learning to improvise, arpeggios are a good place to start. Even better is mixing arpeggios with other techniques. For example, Jamey Aebersold has great patterns for beginning improvisers where you play up the arpeggio (1-3-5-7) and then down the scale/mode. For example, over
Dmin (the ii chord), you could play:
F (1-3-5-7-6-5-4-3) or
But you can also reverse the order: descend down the arpeggio and then play up the scale. Pairing two techniques in this way opens a lot of doors for creative practice.
Your question asks: "If I am in
C Ionian mode, my progression is
CM or any variations of these basic chords." But there's an issue with how you've phrased this. The chords
CMaj aren't in
C ionian--they're in the key of
C Major. Only the
CMaj chord pairs with
C Ionian. The
Dmin chord pairs with
D Dorian minor and the
G7 chord pairs with
G Mixolydian. Referencing these scales (Dorian minor, Mixolydian, and Ionian) is a modal approach because you are drawing on three modes that all come from the same
C Maj parent scale. It would be weird, though, to play the entire progression thinking only of a
C major scale. To illustrate why this sounds weird, go to your piano, and in your left hand play a
Dmin chord while playing a
C major arpeggio in your right hand. Improvisation sounds good when the chord tones are emphasized. This is not possible when thinking of the entire ii-V-I progression as being
C major. Rather, it will sound better to think in terms of the specific chords/modes belonging to each measure.
Instead of playing a major key, we could play in a minor key. For example, instead of playing a ii-V-I in
C Maj, you could play a ii-V-i progression in the key of
C minor. Now, the chords would be
Cmin where the first chord is a half-diminished chord (flatted 3rd, flatted 5th, and flatted 7th) and the second chord has alterations on the 9th, 5th, and/or 13th. There are a few choices of scales to use over this progression, but one nice set of scales is:
D locrian #2 (the 6th mode of the
F melodic minor parent scale),
G altered (the 7th mode of the
Ab melodic minor parent scale), and
C melodic minor. This is again a modal approach because, for each chord, we're pairing a scale that starts on the root of that chord. But this time, the three modes aren't all coming from the same parent scale.
Hopefully these examples elucidate what a mode is, how it is used, and why it is used. That said, when you're improvising, it's really helpful to think about groups of notes. (This sort of technique is called "chunking" in psychology and allows us to store more information using less working memory.) Thinking about the scale in terms of the chord tones can be really beneficial and can have two positive effects: (a) this can help you emphasize the strong tones from the underlying scale and (b) this can help you reduce the amount of working memory you must expend when thinking about which notes to play.