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?