1. Do it with other people
One of the most important aspects of jazz is that it is best done with other people. Jazz involves improvised interaction in real-time between musicians. It is a wonderful and powerful skill to have. You can’t practice this kind of responsive interaction by yourself. That makes playing in an ensemble a must. As well as good music lessons, you need experience playing in a jazz band;not a big band or “stage band”, but a small, improvising group. Big bands are important too, and great fun, but they are focused on different things and teach different skills.
2. Listen to jazz and play by ear
If you are going to play jazz, which is largely based around improvisation, you need to listen to it being played so as to attune your ears to the nuance and style of the musical language. Listen to recordings of it and go to live performances too. This will also teach you what your instrument should sound like, in the hands of a professional. You should also practice playing by ear. Most of what you do in jazz relies on your ears and your ability to imagine sound, not upon your eyes and reading music. Of course you should be able to read too, but that won’t turn you into a good jazz player. Listening and training your ears will. If you’d like to get good at playing jazz, you will also need to practice both technique (how to make your instrument work, and how to control is so as to express yourself) AND improvisation (“soloing” and the skills and techniques that jazz improvisers use). Jazz playing is not just “from the heart”, and is not just something that “you just have” or “don’t have”. Good jazz players work at it. It isn’t magic, it is the culmination of practice and experience over time.
People you hear who are good at playing jazz have practiced doing it. That means practicing the techniques that go into playing such as scales, arpeggios, tone production, sequences, register, endurance, touch, time, and so on. It also means applying those techniques in a structured way to practice improvising, interpreting melodies in different styles, ensemble interaction, etc. Forget the myths about it all just being “from the heart” or entirely spontaneous: that’s just publicity. Like just about everything else you can do, success requires a lot of hard work over time. Even if, in the end, it turns out that your “heart” or “feeling” isn’t as “great” as Miles Davis’s was, you’ll never find out if you can’t play your instrument and don’t know your stuff. And anyway, who cares? There was only one Miles, just as there will only be one you. So get busy and enjoy the journey because there’s nothing else like it!