As far as I know there are basically two formulations of Taylor Expansion.
1) Basic Expansion (linear)
$$f(\boldsymbol{X}) \text{ around } \boldsymbol{x_{0}} \approx k_{0} + k_{1} \cdot (\boldsymbol{X} - \boldsymbol{x_{0}}) \hspace{5 mm} : \hspace{5mm} k_{0}=\frac{1}{0!} \cdot f(\boldsymbol{x_{0}}) \text{ and } k_{1} = \frac{1}{1!} \cdot \bigg(\nabla f(\boldsymbol{X}) \,\bigg|_{x} \bigg)^{T} $$
1) Advanced Expansion (linear)
$$f(\boldsymbol{X} + \Delta x) \approx k_{0} + k_{1} \cdot (\boldsymbol{\Delta x}) \hspace{5 mm} : \hspace{5mm} k_{0}=\frac{1}{0!} \cdot f(\boldsymbol{X}) \text{ and } k_{1} = \frac{1}{1!} \cdot \bigg(\nabla f(\boldsymbol{X}) \bigg)^{T} $$
So you'd think I'd be all set, happily expanding my $V(S,t)$ and $V(S + \Delta S, t + \Delta t)$ ever after. But I am reading on Explicit Finite Difference method to solve PDEs and it is full of hybrids like $V(S,t-\Delta t)$, $V(S+\Delta S,t)$ and $V(S-\Delta S,t)$ inducing sounds of crackling popcorn in my head. How do I expand them (and I'm not even touching quadratics)?
This is all you need, especially with the given example.
Thus, for instance, $$V(S \pm \Delta S, t) = V(S, t) \pm \partial_S V\Big \vert_{(S, t)} \Delta S + \partial_S^2 V\Big \vert_{(S, t)} \big(\Delta S\big)^2 + \mathcal{O}\Big( \big(\Delta S\big)^3 \Big)$$