Of course the melody is important but: the harmonic framework is as important and always the backbone of a standard. Without it the melody just "hangs in the air". 

The STEPS to follow:

  1. Get to know the MELODY: Play along/sing along (2x) MEMORIZE it, and after that TRANSPOSE it to another key
  2. Get to know the BASS-LINE: Play along/sing along (2x) 
  3. Get to know the CHORDS: Play the chords on any keyboard instrument (from piano to melodica): close position in the right hand. Don't worry about voice leading, just grab each chord.  In that way you learn to understand each chord as a whole.
  4. Get to know the chords by making ARPEGGIOS: use the given example to begin with, and when you are fluent: make your own!
  5. Learn to play the simplified CHORD SCHEME on a keyboard instrument (bass note left hand, three additional notes in the right hand). Pay close attention to the way chords are linked to each other (voice leading) be aware of small "lines" in different voices. Start slowly, speed up when possible. (Guitar players check the material of Leon Mennen!).
  6. Back to the MELODY: if there are notes in the melody that are not part of the notated chord symbol... encircle them. What do they add to the chord symbol of that moment?