How do you go from the 4-vector to the stuff about direction. I am sure I did the boost correctly, I am just lost on how to proceed. I’ve been out site since school started so I’ve been trying to play catch up.

I think you’re asking about how to relate Cartesian and spherical coordinates.

Our formula for Lorentz boosts was written in Cartesian coordinates. What we’re interested in is the relation between the corresponding systems of spherical coordinates.