> While it's almost always I, IV, V, and vi, we have both II/ii and III/iii, differing on whether the third is major or minor.
This might be a bit confusing as written, because the slash "/" is normally used in roman numerals to indicate secondary chords. For example, the II chord may be written as V/V. (V/V is always II, but not the other way around... the "/" notation indicates that the chord is functioning as a secondary chord, which is something you figure out from context.)
The terminology "open chords" is also a bit unusual. The term "open chords" usually refers to chords that use open strings (strings that are not fingered). Chords without a third are often called "power chords". You can do your entire song with power chords if you're playing rock music.
I'd also add that what chords you get as the three/four chords you need depends so heavily on genre. If you were playing blues, you might pick I7, IV7, and V7 for your first three, and then maybe bVII7 as the fourth. If you were playing jazz, you might pick something like I69, ii9, IVmaj7, V7.