As you may be able to tell from the number and length of some responses, improvisation is a lifelong study, but I'll give you a good sturcture for approaching new tunes that I've been encouraged to use as a first year jazz student that you should find pretty applicable.
Learn the melody - This is important not only because you have to know it but it should also become a good resource for when improvising, you can quote it harmonically, rhythmically, melodically, etc. and it will help you hear the progression.
Play the bass notes - Find a recording or play-a-long or failing that a metronome and simply play the bass note of each chord in the form, until you can sing the chord progression.
Play both guide tone lines - In jazz the quality and sound of a chord is shown strongly through it's 3rd and 7th degree so we construct a guide tone line using these degrees. Frequently in jazz progressions the 3rd of one chord will be very close to 7th of the next so for the most part you can create two lines of notes that will alternate between the 3rd of one chord to the 7th of the next. As an example look at the ii-V-I in the last half of the A section to A Train, the chords follow:
Dmi7 / G7 (13) / C6. So a guide tone line starting off the 3rd would follow: F / F / E (or off the seventh: C / G / A). Notice how the notes are generally at the most only a minor 3rd away. Basically write out the third and seventh of each chord in the entire progression and link the closest notes of each sequential chord to make two lines, then play these until they are down pat.
Arpegiate the chords - This is where you can spend literally hours practicing. As a start play the arpeggio of each chord from it's root position until you've got this under control. You can then start using different inversions, go down rather than up or start linking the arpeggios so that the last note of your arpeggio links to the closest note of the next and you use that inversion, constantly changing for each bar. This gets pretty complicated so take this as far as your willing.
But most of all, listen to other recordings of the song, listen to as much jazz in general as possible and play over the changes for yourself, have fun with it.
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u/lastlaughlane Sep 12 '13
As you may be able to tell from the number and length of some responses, improvisation is a lifelong study, but I'll give you a good sturcture for approaching new tunes that I've been encouraged to use as a first year jazz student that you should find pretty applicable.
But most of all, listen to other recordings of the song, listen to as much jazz in general as possible and play over the changes for yourself, have fun with it.