For learning tunes, there’s a few steps to take. There’s no magic pill or anything special about it, it is just hard work. But you can learn jazz standards a little easier with these steps.
Learn the melody
Find the tune in a RealBook or google for it and you’ll find the sheet music pretty easily. First thing is to simply go through and learn the basic melody and be able to play it by memory. Next, listen to a few recordings of the tune and hear how others are playing the melody. If you like how someone does a rhythm differently or a little lick they play within the melody, learn what they’re doing and apply it to you’re interpretation.
Learn the chords
Start simple and get the basic triads down. Run the triad in a slow tempo through the entire tune. Then, do the same and add any extensions such as the 7th, b9, etc. Be able to do this by memory as well.
If you have a smart phone and have the iReal App (http://www.irealb.com/), this is a great tool to learning the chords as you can have it loop 4 measures, 8 measures, etc until you’ve learned the entire tune.
Next, run the chords up and down in various patterns and as you approach the next chord, start that chord at the closest chord tone to the one you were just on. For example, if you are moving from F-7 to Bb7 and on beat 4 you are playing a C on the F-7, start the Bb7 either on D or Bb (the 3rd or root) and run that chord.
This is a very important step and you should take your time with it. Really dig into the sound of the tune and the sound of the chords. Be able to run up and down the chords without hesitation and really being able to anticipate the next chord. This is very specific practice and you’ll find when you are later in performance, you’ll naturally and subconsciously start hitting the cool tension and release notes.
Also, if you have access to a piano, learn the chords this way. This will give you a good visual and you’ll start to see how one chord relates to the next one, how the notes line up and you’ll see the natural tendencies of which notes want to resolve to the next ones.
Put it all together
Using either the iReal App or a play-along CD or playing with friends, play the tune as if you’d be performing it. The previous steps are very specific practice and not intended to be necessarily the time for being an artist so-to-speak. This is the technical/mental aspects of practicing. Later, when you are performing, be free, be an artist and play from your heart.