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 $Y$).
EDIT: Just to give an example of what things can occur, consider the singularity $k[x,y,z]/(x^n + y^n + z^n)$ for $n \geq 2$. Then blowing up the origin resolves the singularities but the relative canonical is divisor $K_Z - p^*K_X = (2-n)E$. In particular, every coefficient $\leq 0$ occurs. If you are willing to look at higher dimensions, then every integer appears as a coefficient.
By the way, if $X$ is not Gorenstein (or quasi-Gorenstein/1-Gorenstein), then $p^* K_X$ can be substantially harder to understand. For some different definitions see the paper of de Fernex and Hacon, Singularities on normal varieties.
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.