I need to determine what shape the surface is and justify it.
Now wolfram alpha tells me that this particular surface is a hyperbolic paraboloid which has the general form: $\alpha x^2-\beta y^2-z=0$ but when I try to manipulate my equation by completing the square the closest I can get is this: $(x+y)^2-(x+z)^2+x^2-2yz-y=0$ which isn't quite right. Can anyone point me in the right direction as to what I should do next?
Don't guess, just diagonalize the quadratic form. Ignoring the linear terms (which can be dealt width by moving the origin), the quadratic terms can be written as $$\vec{r}\begin{bmatrix}1&1&-1\\1&1&-1\\-1&-1&-1\end{bmatrix}\vec{r}$$ where $\vec{r}=(x,y,z)$. In the eigenframe, the signs of the eigenvalues tell you what kind of surface you have, and the eigenvectors tell you the orientation of the principal axes (which tells you which linear combinations you need to complete the squares.