Learning a Jazz Tune in a Week:
Day 1: Pick the tune you want to learn and find a recording that you like of it. A good place to start is usually with the artist that first recorded the tune. Frank Sinatra is also a good choice if he has recorded it. He tends to not embellish the melody a lot. Chet Baker is another person who keeps it simple.
Day 2: Now that you have a tune picked learn to sing it. If it has words learn the words. I would not worry about learning the solos just the tune.
Day 3: Try to put the tune on your instrument at this time. Take your time with this process it my take longer than you think. Once you think you have the tune play it and record it. Does it sound like the recording or is it different. The goal is to make it sound like the recording as much as possible. Once you have done that check your work against a Jazz Real Book. Usually you can search and find one online.
Day 4: Try and play the melody starting on a different note. If possible try to figure it out in all twelve keys.
Day 5: Try to put the tune over a backing track that you find on Youtube. Most popular standards have one. Try listen and can tell when the chord (harmony changes).
Day 6: Try using the backing track to play the melody then use the track to change the melody and make it you own version. Play the melody as written one last time.
Day 7: Take a break.
A quick note: do not pick a hard song to start with, it is a process that takes time to work through. Start simple and build. St James Infirmary is a great first tune. I recommend the Louis Armstrong recording. If you want to go to easier tunes first I will also link some basic tunes to learn by ear as well.
List of easy tunes to learn by ear.