What is the correct order when multiplying both sides of an equation by matrix inverses?

You must be sure to multiply on the correct side. To get rid of the $B$ in $BAC$, you must multiply on the left by $B^{-1}$, so you must do the same on the righthand side of the equation:

$$AC=B^{-1}BAC=B^{-1}D\;.$$

To get rid of the $C$ in $AC$, you must multiply $AC$ on the right by $C^{-1}$, so you must do the same thing on the other side of the equation:

$$A=ACC^{-1}=B^{-1}DC^{-1}\;.$$

You have to multiply $C^{-1}$ from the right ( on both sides) and $B^{-1}$ from the left ( on both sides). Which one you do first doesn't matter.

To get from $BAC = D$, you left-multiply both sides by $B^{-1}$ and right-multiply both sides by $C^{-1}$, giving $B^{-1}BACC^{-1} = B^{-1}DC^{-1}$. Since $B^{-1}B = I$ and $CC^{-1} = I$, this simplifies to $A = B^{-1}DC^{-1}$.