A Universe of Sorts

"oompah" bass with chords on beats two and four
chromatic mediants
augmented sixth chords (a fan of Joplin will see them everywhere)
heavy right hand syncopation in counterpoint to steady left hand
tuneful melodies
straightforward forms: binary (simple / rounded), ternary, rondo, etc.
tempi typically vary from Andante to Allegro, hardly anything much slower or faster. If faster, you end up in "stride" piano territory, which is not what you are trying to accomplish.

Music.se link
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 rising 4th intervals onto the tonic, e.g., in C, in its full extension: C - E - A - D - G - C.

The fifth of each chord in a Major key signature can be played as a preceding chord that basically builds tension and release when resolving to the diatonic chord of the given key signature.

A secondary dominant is a dominant chord that resolves to any one chord in the scale, other than the tonic. So V/V is a secondary dominant, but so is V7/IV and so is V/ii etc.

Music Matters YT video so we play a V-I in a different key. We need the I of the secondary dominant to be a chord to the key we are in.

There isn’t a singular secondary dominant for C major. There is a respective secondary dominant for every note in the scale (besides C since that’s the normal dominant). So D major (II) is the secondary dominant of G major (V), E major (III) is the secondary dominant of A minor (vi), F# major (#IV) is the secondary dominant of B diminished (viio) etc.

The V/V chord: The V/V chord is the "five of five" chord. Ask yourself this question: If I switch to the key of the V chord, what is the V chord in that key?" So instead of playing V, play V/V.
C - C7/Bb - F/A - Fm/Ab - C/G - G - C from Joplin's the entertainer.

Reddit thread
Ragtime uses fairly fancy European chord progressions. The usual primary chords, secondary dominants (sometimes in a chain), diminished 7ths, Neapolitan and augmented 6ths, etc.

Came from operatic composers, from between baroque and classical era.
Traditionally used in minor. Is a predominant chord. Comes before V7.
It is a flat 2major triad in first inversion.
For example, in G minor (Bb major). Flat second is Ab. We The Ab major triad is Ab C Eb. First inversion means that the third is in the bass, so we play C Eb Ab'.
Can play N6 V7 i.

One that I find very "signature ragtime" is the cadential progression V7/V - V7 - I. For example, in C, that would be D7 - G7 - C, with a chromatic D7 rather than the diatonic Dm7.

Try this one. In C, C7, F, F#dim7, G7, A7, D7, G7, C
In Joplin's "School of Ragtime", he gives all the exercises IV - viiº7/V - I64 - V7 - I
(In C: F - F#dim7 - C/G - G7 - C).
This is a pretty generic 19th-century cadence; it's also the very first chord progression shown in Czerny's "Art of Preluding".

Recall the notation of /: C/G is a slash chord, C is the main chord, so we play the chord C E G with the right hand. The slash means "over top of". So in the left hand, we play the G note. (we play the chord on top of the bass note G).
Recall what viiº7/V means: We want to play the viiº7 chord,

The chord viiº7/V is a diminished seventh chord built on the seventh scale degree of the dominant (V) of a key. This is known as a secondary leading-tone chord. It functions as a leading-tone chord that resolves to the dominant (V) chord, just as a viiº7 chord in a key resolves to the tonic.

§ Ragtime Composition