The product of two (independent) Gaussian random variables is not necessarily a Gaussian random variable! (See here).

Here are some results:

Let $X:=(X_1,\ldots,X_n) \sim N(\mu,C)$ jointly normal.

- Let $b \in \mathbb{R}^m$, $A \in \mathbb{R}^{m \times n}$. Then $$Y:=(Y_1,\ldots,Y_m) := A \cdot X+b$$ is normal, precisely $$Y \sim N(b+A \cdot \mu, A \cdot C \cdot A^T)$$
- (Special case of 1.) $$Y:=\ell^T \cdot X = \sum_{i=1}^n \ell_i \cdot X_i$$ is normal where $\ell \in \mathbb{R}^n$ arbritary, precisely $$Y \sim N(\ell^T \cdot \mu, \ell^T \cdot C \cdot \ell)$$ In particular ($n=2$): $a \cdot X_1 + b \cdot X_2$ is normal for all $a,b \in \mathbb{R}$.

Now let $X,Y$ normal and independent random variables. Then $(X,Y)$ is a normal random variable (so you can apply 1. and 2. to $(X_1,X_2):=(X,Y)$).

Since you asked in particular for functions $f: \mathbb{R}^2 \to \mathbb{R}$: Let $$f(x,y) := a \cdot x + b \cdot y + c \qquad (a,b,c \in \mathbb{R})$$ then $f(X,Y)$ is Gaussian.