I am a clarinetist/saxophonist also getting into jazz, but most of the same things will apply.
Much of jazz is based on "standards" or common tunes that people play. You can pick up a "real book", which is like the jazz standard bible, or you can find a PDF online of an older version. Certain Places will have their own set of standards which they like to call and know well. Many jazz musicians know these tunes, the melody and the chords and all. If they don't know a tune, they can learn it by ear or check something like a real book. This includes the chord progression, key, melody, etc.
That's how they learn tunes, but that's not how they play them. Jazz, like any form of music, has stock skills. Scales, chords, arpeggios, licks, that sort of stuff. A lot of jazz musicians practice these on multiple keys, so that no mater what they are playing, they have the skills to play. So, most of the groove type of stuff is often just taking those stock skills and applying them to whatever chord progression the group is playing (it's not always what is on the lead sheet, so watch out). It seems magical, but after enough practice, these things become automatic.
So, just like anything, practice you scales and chords, learn your theory, TRAIN YOUR EARS, and most importantly, play. Play a lot. You will pick up a lot of things like groove and improvisation as you play more and more. Your time in the practice room just gives you mechanically keep up and perform. The experience of playing teaches you how to play music.