Let's say we have an equation and we want to transform it by the matrix $A$, where:
$$\begin{pmatrix} x' \\ y' \end{pmatrix} = A \begin{pmatrix} x \\ y \end{pmatrix}$$
Now, everywhere I've seen has said to find the $x',y'$ vector and substitute the resultant formulas for $x'$ and $y'$ into the original equation. However, I've found I only get the inverse transformation by doing this. Am I correct in thinking I have to find $x$ and $y$ in terms of $x'$ and $y'$ instead?
You are correct. You have to substitute something for $x$ and $y$ to transform the equation, so you need to express these old variables in terms of the new variables $x'$ and $y'$. Doing that entails inverting $A$.