Familiarize yourself with more complex chord structures. Jazz hardly ever uses triads (chords built with three notes). Jazz chords typically feature chords of four notes or more, adding the diatonic seventh of a scale to the basic triad. Jazz also incorporates notes beyond the seventh. These are called upper extensions and create ninth and eleventh chords.
Refer to chord progression by position in the scale rather than by actual chord names. This makes it easy to memorize progressions in any key. Many jazz musicians improvise their parts rather than play written music. This means calling out a key and then indicating a progression within that key. An example of this would be Imaj7, IV7, and V7. Regardless of key, this tells you to play the major seventh built on the first note of the scale, the seventh built on the fourth note of the scale, and the seventh built on the fifth note of the scale. These positions use Roman numerals as indicators.
Learn how to read and count complex time signatures. Many jazz songs are written in odd time signatures such as 5/4, 7/8 and 11/4. Common time is 4/4, which means 4 beats to a bar. Reading music in odd time signatures means you will need to acclimate yourself to counting in rhythms that may feel unnatural to you at first because the main beats are placed at odd points throughout many jazz pieces.
Practice playing with jazz fake charts. Since jazz music is often improvised, learning to make up musical passages over a set of chords and lyrics is a good way to build your jazz playing and reading skills. It's not uncommon to walk into a jazz rehearsal where you're handed a set of chords and lyrics and told to play.