Following is a simple outline for a basic approach to learning jazz improvisation. I can't take credit for these ideas because they are common to any decent jazz teacher, and are described in similar ways in many method books and textbooks. I am posting it simply to help prospective students understand the method I typically use to teach melodic improvisation through the language of jazz...
Beginner Level
1. Embellish the melodies of songs by ear (using both rhythm and pitch variations).
2. Improvise on major/minor scales by ear – freely at first, then in the context of applicable songs (specifically, ones that do not contain changes in tonality or chords that go outside the key).
Intermediate Level
1. Improvise on the chord tones of various songs (7th chords primarily; 6th chords and/or triads when applicable) – write out the chord tones; then practice playing/singing through chords only; then practice making melodies using chord tones only.
2. Improvise on the chord tones of songs and add "connecting tones" by ear.
Advanced Level
1. Improvise on chord tones, including upper extensions (9, 11, 13), alterations (b5, #5) and altered extensions (b9, #9, #11, b13) – use the same process as described above for 7th chords.
2. Write out scales associated with each chord, or groups of chords when applicable, and improvise on them.
3. Study and practice common “chord scales” – separately at first, then while improvising on songs.
Pro Level
1. Improvise using nonharmonic tones associated with each chord scale – practice using each category of nonharmonic tones separately at first (including passing tones, neighboring tones, appoggiaturas, escape tones, free tones, chromatic approach tones, chromatic approach tones with leaps, double chromatic approach tones, double chromatic approach tones with leaps, changing tones, chromatic sequences); then practice using them freely.
2. Practice advanced “chord scale” concepts – separately at first, then within various harmonic contexts.
3. Explore advanced jazz improvisation concepts, such as motivic development, intervallic improv, superimposed harmonic structures, "outside" playing, advanced rhythmic concepts, free improv, etc.
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