To expand upon those two chords we’ve talked about, some of those “other” chords are used differently. The ii, IV and vi are chords called Pre-dominant or Intermediate harmony. Simply put, these chords go to V or V7, they form a bridge from tonic to dominant. In some cases, like the deceptive cadence a piece of intermediate harmony can follow a dominant chord.
With the use of intermediate harmony, we can explore two of the most common cadential formulae in western music. The first is the ii V I progression which pervades jazz music in addition to classical. The root motion of falling fifths or ascending fourths that was mentioned in the tonic and dominant discussion makes ii V I progression very strong. Here’s what it looks like in notation (key of C).
The other common progression is IV V7 I.
If you want to explore this concept further, the best bet is to write out a few ii V7 I or IV V7 I progressions in various keys. The sharp keys, G, D, A and E, will probably work the best to start with.