Structurally ragtime harmony is pretty much classical tonal harmony, but there are of course some idiomatic specificities that give ragtime its characteristic sound.
One progression very characteristic of ragtime is the so called... ragtime progression (although it was used before, even in classical music, it was mostly popularized in ragtime). It's made of a succession of "piggybacked" secondary dominants in a succession of 4th intervals onto the tonic, e.g., in C, in its full extension:
C - E - A - D - G - C (in this instance I find chord note names much easier to use than the farfetched roman numeral analysis that would be required)
On one variation, all chords are major and usually 7th dominant chords. Parts of the progression may be omitted (i.e. starting on the A or the D), the most usual is a 4 step progression (in this case, starting on the A).
Another variation superimposes a ii-V-I progression by making one of chords minor (also usually 7th)
C - A - Dm - G - C
There is also a lot of chromatic substitutions, usually dictated by voice leading (i.e. chords changing by step intervals, mostly half tones) thus creating a lot of full and half diminished (7th) chords.
Just to give an example, at the end of the B part Scott Joplin's Maple Leaf Rag (Ab major), the progression is:
Notice how the first chord of the "ragtime progression", F in this case, is approached chromatically in a quite seamless way. Then the progression follows pretty much plainly (no no-chord-tones for two bars).
In bar 4, at the point marked with *, Bbmin is replaced by Bb7add9 both providing harmonic interest and smoothing voice leading: notice how the uppermost and lower "treble" voices move stepwise.
Regarding scales, my understanding is that structurally they are mostly diatonic, I don't think that pentatonic or blues scales are inherently used, although sometimes a "bluesy" sound may occur by means of the harmony. Melody is not a strong component of ragtime anyway, many ragtimes almost have no melody as such, or melody is built practically with arpeggiated chords and a lot of chromatic passing tones (Mapleleaf Rag a case in point).