Mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$?

We can avoid using the fact that $X^2\sim\sigma^2\chi_1^2$, where $\chi_1^2$ is the chi-squared distribution with $1$ degree of freedom, and calculate the expected value and the variance just using the definition. We have that $$ \operatorname E X^2=\operatorname{Var}X=\sigma^2 $$ since $\operatorname EX=0$ (see here).

Also, $$ \operatorname{Var}X^2=\operatorname EX^4-(\operatorname EX^2)^2. $$ The fourth moment $\operatorname EX^4$ is equal to $3\sigma^4$ (see here). Hence, $$ \operatorname{Var}X^2=3\sigma^4-\sigma^4=2\sigma^4. $$

Note that $X^2 \sim \sigma^2 \chi^2_1$ where $\chi^2_1$ is the Chi-squared distribution with 1 degree of freedom. Since $E[\chi^2_1] = 1, \text{Var}[\chi^2_1] = 2$ we have $E[X^2] = \sigma^2, \text{Var}[X^2] = 2 \sigma^4$.