If $X$ is smooth and you are blowing up a smooth subvariety $Y$ of codimension $n$, then $K_Z = p^*K_X + (n-1)E$.  This is an exercise in Chapter II, Section 8 of Hartshorne.  

If $X$ is not smooth at least at the generic point of $Y$, then I don't think much can be said.  If $X$ is smooth at the generic point of $Y$ though, then at least one of the components of $K_Z - p^*K_X$ will be $(n-1)E$ (that will be the component dominating $Y$).

Can I ask a dumb question, what does it mean to be *normally flat*?

**EDIT:** Nevermind, I answered my own question.  Normal flatness is a condition in Hironaka's proof of resolution of singularities.  I knew I had heard it somewhere before.