Morse's style incorporates a hell of a lot of influences, and on top of that, there's a lot of serious academic knowledge informing his music. I'd say focus on the following.
Alternate picking: This is really a given, especially cross-picking arpeggios.
Chicken-picking, open-string scales and steel-guitar bends: Maybe get some country guitar instructionals?
Harmonics: Learn the locations of each harmonic along the strings, and practice harp-harmonics, Lenny Breau-style.
Classical guitar: Brush up on your classical fingerpicking. Maybe find some Carulli or Giulani pieces of something to get you started?
Vibrato: Develop a few different ones, as Morse has a lot of control over this area of his playing. It ranges from a subtle classical vibrato to an insanely wide, fast one that sounds like a whammy bar.
Chromatic notes: Get used to incorporating chromatic passing notes into your soloing.
Pick-up switching: Get used to switching pick-ups on the fly while you play, as this is a key part of his sound. You might even want to invest in a coil-tap switch for your guitar if you don't have one.
That's the technical aspect out of the way. In terms of theory, I'd say learn as much about harmony and counterpoint as you can (Bach's music is an especially big influence on him), as this is a key part to his writing, and try and absorb the specifics of as many different genres as you can. Also, train your ears, as Morse has developed them to the point where he was able to jam on stage with Vinnie Moore and repeat any of Vinnie's licks note-for-note on the fly.
In terms of gear, multiple PU options on your guitar are a must, as I've said, and setting aside his amp (he's using ENGLs these days, and has used 5150s (in Deep Purple to blend better with the organ), Marshalls and Ampegs in the past), delay is a huge part of his sound, so a high quality delay set up is a must. Nowadays he uses two Memory Man units, one set of a long delay, and the other for a short one, and has them permenantly turned on, using a pair of volume pedals to fade them in and out, so if you could come up with an arrangement like that (I'm pretty sure the POD XT and X3 allow you to use the expression pedal to control delay level) it would be ideal.
I know that's a lot, but studying everything about Morse could easily be a lifetime's pursuit!