When I first started playing around with jazz it was really by being flung in at the deep end by a lecturer that taught me guitar at uni, when he asked me to play with his band at a gig when I was in the audience.
I got up, the clarinetist said "there's a chord chart, it's blues with more notes, you'll be fine" and we were off. Since then, that's pretty much how I approached jazz, chord wise, which obviously isn't something that many purists would dig, but I found it an easy way in to the general vibe of things, note choice and voicing wise. Started playing 12 bars with walking bass lines and forcing myself to play a new chord (or inversion) on every beat with extra melody notes amongst them. After a while playing around with those ideas I started actually learning playing around with the more common jazz progression which was easier than I think it would have been, since I already had a few bits and bobs I could use that I'd adapted from what I was already familiar with.
Alternatively, learn some standards and play around with voicings. Generally, knowing lots of voicings and inversions gives you plenty of room to spice up chord progressions and give them a jazzy vibe, to the less discerning ear, anyway :lol: