When I learn a new jazz standard, I like to practice the changes in chunks. I try to isolate anything I recognize from other tunes or concepts and then put the pieces together. It's a lot like learning to read; you learn the letters first, then recognize words without thinking about the letters, then recognize phrases without thinking about the words.
The primary building block is learning chord types. I regularly practice chords, arpeggios, scales, and improvisation over maj7, 7, m7, and m7b5 chords separately.
The next step is learning commonly occurring mini-progressions. I've been working on iim7-V7, V7-Imaj7, Imaj7-IVmaj7, I7-IV7, iim7b5-V7b9. After that, iim7-V7-Imaj7 follows easily. When I'm ready, I'll expand to larger progressions through the cycle of fourths, like iiim7b5-VI7b9-iim7-V7-Imaj7. Should be cake with a solid foundation.
The goal is to get faster at recognizing these common chunks, comping or soloing over them without thinking about every chord. I hear a lot of players talk about simplifying their thinking to accomplish the same thing, playing a single major scale over an entire iim7-V7-Imaj7-vim7 progression for example. I'd rather start with the basics, looking with more detail, and get so accustomed to outlining the subtleties of each chord that I can rely on the thinking I've done in the past to come through my fingers automatically.