First of all, the predominant feature of jazz is the swing feel. In most forms of jazz (swing, bebop, hard bop, east coast, west coast, dixieland, etc...), the eighth notes have a swing feel, where the down beats have a longer duration than the up-beats. That's what drives the music forward.

Playing with a "swing" feel almost instantly makes you sound more jazz. Play a simple major scale in swung eighth notes, and it will sound jazzey.

Now, you said you were confused about how jazz goes through so many key centers, and such... Here's a few things to look up and research (they should reveal a few things about why jazz goes through different keys) -

secondary dominants, tritone substitution, chordal extensions, qualities, chord substitution, dissonance, scale substitutions, rhythm changes.

I would suggest going to jazzbooks.com and getting books such as "The Scale Syllabus," "Gettin' it together," "Major and Minor," and "Volume 1 - anyone can improvise."

A lot of times you'll find that the scales that are played are in someway related to the chords they are played over. As said before, chromatic passing tones are used to a great extent. Dissonance also plays a huge part - sometimes a minor third will be played over a major chord, in order to produce a desired "sound." Listen to a lot of jazz, buy a fake book, listen to the solos, and find out why they do the things they do. Slowly but surely, jazz will become demistified.