How to prove $\mathbb Z_3\rtimes(\mathbb Z_2\mathbb \times\mathbb Z_2) \cong S_3\times\mathbb Z_2$? I know that there is a unique semidirect product $\mathbb Z_3\rtimes(\mathbb Z_2\times \mathbb Z_2)$, defined by mapping two of the order two generators of $\mathbb Z_2\times \mathbb Z_2$ to the inversion automorphism of $\mathbb Z_3$. 

However, I am not exactly sure how to proceed to show that $\mathbb Z_3\rtimes(\mathbb Z_2\mathbb \times\mathbb Z_2) \cong S_3\times\mathbb  Z_2$.

What exactly is the map I could construct?
Let $G = S_3 \times \{\pm 1\}$ (writing the cyclic group multiplicatively). 

Now map 
$$
G \to \{\pm 1\} \times \{\pm 1\}
$$
by the rule
$$
(\sigma, \epsilon) \mapsto (\epsilon \cdot \text{sign}(\sigma), \epsilon). 
$$
The kernel is $A_3 \times \{1\}$, so we have an exact sequence
$$
1 \to A_3 \to G \to \{\pm 1\} \times \{\pm 1\} \to 1,
$$
and there is a splitting $\{\pm 1\} \times \{\pm 1\} \to G$ given by $(-1, 1) \mapsto ((12), 1)$ and $(1, -1) \mapsto (\text{id}, -1)$.

It follows that $G$ is some semidirect product of $A_3$ by $\{\pm 1\} \times \{\pm 1\}$ as desired (using $A_3 = \mathbb{Z}/3\mathbb{Z}$). We still must check that it is the non-trivial semi-direct product, i.e. that the lift $((12), 1)$ acts non-trivially on $A_3$ by conjugation. This is indeed true since e.g. $(12)(123)(12) \neq (123)$.
If $G$ is the group defined by the semidirect product you defined and $H$ is the direct product of $S_3\times \mathbb Z_2$, then all elements of $G$ take the form
$$g=(x_1,x_2,x_3)$$
where $x_1\in\mathbb Z_3$ and $x_2,x_3\in\mathbb Z_2$. All elements of $H$ take the form
$$h=(y_1,y_2)$$
where $y_1\in S_3$ and $y_2\in\mathbb Z_2$.

If I understand your definition correctly, then for the group $G$, the following should hold:
$$(x_1,0,0)(a,b,c)=(x_1+a,b,c)$$
$$(x_1,1,1)(a,b,c)=(x_1+a,b+1,c+1)$$
$$(x_1,0,1)(a,b,c)=(-x_1+a,b,c+1)$$
$$(x_1,1,0)(a,b,c)=(-x_1+a,b+1,c)$$

Then you should be able to find an isomorphism between $G$ and $H$ by mapping the elements of $G$ that take the form $(x_1,0,0)$ and $(x_1,1,1)$ to elements of $H$ that take the form $(y_1,0)$ and elements of $G$ in the form $(x_1,0,1)$ and $(x_1,1,0)$ to elements of $H$ in the form $(y_1,1)$.