For jazz experts or longtime jazz players: What should I master early on?

learn chord changes.  you may get a lead sheet every so often with actual parts, but chord changes is where you'll live.  some charts will have the lead with chords written on them to help you with the rhythm.  

after that, listening.  i would just start listening to a bunch of jazz guitarists.  nothing wrong with following along and "stealing" other people's work and licks until you find your own style.  

there are apps where you can play off chord changes alone on iphone that don't cost a lot where you can play along with jazz classics to work on improvisation.  iReal Pro is one I use.  but look around and see what suits you best.

next step is just to get in a band and start playing.  that is where real experience comes in and there is nothing like playing with others for critique, playing with eye contact, and just hearing other live players.  if you're out of school, look at local community colleges if they have small group jazz or jazz combos or big band, depending on your preference.  try them all.  it may cost a little money to sign up, as they would be a class, but you'll learn and have fun.  look at your local schools online and see what's available.  

How to swing. Really.

I'm no jazz expert but I'm pretty certain a lot would depend on what instrument you play.