This is a lesson given to a student this week.
Louis Armstrong, on the topic of how to improvise, said something to the effect, "memorize the melody, mess with the melody, and then mess with the mess." For beginners, this is the best advice I've ever come across. It is truly the shortest distance between A and B. Or jazz newbie to intermediate jazz student and beyond.
The student can already play. Therefore, it is a question of what to play and less of how to play it.
Two octave scales: C, F, Db, Bb, Eb
Four note broken chords: C, F, Db, Bb, Eb
Two octave arpeggios: C, F, Bb, Db, Eb
Please use a metronome and practice different tempi, dynamics, and articulations.
Being able to play the broken chords is an important starting point in improvisation.
You've got to learn your instrument. Then, you practice, practice, practice. And then, when you finally get up there on the bandstand, forget all that and just wail.
I'm a professional pianist and music educator in West Toronto Ontario. I'm also a devoted percussionist and drum teacher.