The I, II, and V chords often occur as a II-V-I chord progression, the most common chord progression played in jazz.
... The II chord is always a minor 7th chord, the V chord is always a dominant 7th chord, and the I chord is a major 7th chord.
The II-V-I in the key of F is G-7, C7, FΔ.
V of V
"V of V" means a dominant chord resolving down a 5th to another dominant chord, as in C7, F7.
Sometimes you will see several V chords in a row, following each other counterclockwise around the cycle of fifths.
The quintessential example is the bridge of George Gershwin's "I've Got Rhythm," with four V chords in a row, each resolving down a 5th.
I-VI-II-V
One of the most common chord progressions in jazz is I-VI-II-V.
The original first four chords of George Gershwin's "I've Got Rhythm" are a I-VI-II-V (C, A-7, D-7, G7).
III-VI-II-V
A common variation of I-VI-II-V is III-VI-II-V, a chord progression often used in turnarounds.
In the key of C this would be E-7, A-7, D-7, G7.
I-II-III-IV
I-II-III-IV is often played by pianists and guitarists when a major 7th chord lasts two bars.
The chord in the seventh and eighth bars of Jerome Kern's "All The Things You Are" is just CΔ.
To Provide some contrast and movement, you could ascend I-II-III-IV, and then turn around and descend III-II-I.
I-IV
Major chords are often followed by chords a 4th up.
Sometimes the chord a 4th up will be a major chord, as shown ..., where Bobby Hutcherson follows B♭Δ with E♭Δ in the first bar of Victor Young's "My Foolish Heart."
Sometimes the chord a 4th up following a major chord will be a dominant chord, as in bars 7 and 8 of "Stella By Starlight," where E♭Δ is followed by A♭7.
There's another example in Willard Robinson's "Old Folks," where B♭Δ is followed by E♭7.