I'd like to know how a set of this would look?
$$x_{1}+x_{2}=3$$
$$-x_{1}+2x_{2}=6$$
It's easy to make one set, where you calculate the solutions of this linear system (in this case we have $0$ and $3$). This set would be $S=\left\{0;3\right\}$
How about the other set? I'm very excited to know that, I would say it would be:
$$S=\left\{x_{1},x_{2} \in \mathbb{R}|x_{1}+x_{2}=3 \wedge -x_{1}+2x_{2}=6\right\}$$
Would this be correct or is it written different?
The solution set to the system of linear equations $$x_1 + x_2 = 3 \\ -x_1 + 2x_2=6$$ is the set of all ordered pairs $(x_1, x_2)$ that simultaneously solve both equations. The only such ordered pair in this case would be $(0,3)$, so the solution set would be $\{(0,3)\}$.
To write the solution set out in set-builder notation, you'd write $$\{(a,b)\in\Bbb R^2 \mid a+b=3 \wedge -a+2b=6\}$$