Consider an ARMA(1,1)-GARCH(1,1) model \begin{aligned} x_t &= \varphi_0+\varphi_1 x_{t-1}+\varepsilon_t+\theta_1\varepsilon_{t-1}, \\ \varepsilon_t &= \sigma_t z_t, \\ \sigma_t^2 &= \omega+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2, \\ z_t &\sim i.i.d.(0,1) \end{aligned} for some zero-mean, unit-variance distribution $d$. The model's distributional assumption ($i.i.d.(0,1)$) is on $z_t$, not $\varepsilon_t$. $z_t$ are known as standardized errors (innovations, shocks, errors, residuals). In models where $\sigma_t^2$ is constant (no GARCH part), $\varepsilon_t$ is proportional to $z_t$ (just their variances differ) and you need not standardize but rather can work directly with $\varepsilon_t$.

You examine ACF and PACF of $\hat z_t$* as the i.i.d. assumption rules out nonzero autocorrelation in $z_t$. You examine ACF and PACF of $\hat z_t^2$ as the i.i.d. assumption rules out autoregressive conditional heteroskedasticity in them.

*Or $\hat\varepsilon_t$ if $\sigma_t^2=\sigma^2 \ \forall \ t$ here and further. Hats denote fitted values.

Consider an ARMA(1,1)-GARCH(1,1) model$\color{blue}{^{*}}$ \begin{aligned} x_t &= \varphi_0+\varphi_1 x_{t-1}+\varepsilon_t+\theta_1\varepsilon_{t-1}, \\ \varepsilon_t &= \sigma_t z_t, \\ \sigma_t^2 &= \omega+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2, \\ z_t &\sim i.i.d.(0,1) \end{aligned} for some zero-mean, unit-variance distribution $d$. The model's distributional assumption ($i.i.d.(0,1)$) is on $z_t$, not $\varepsilon_t$. $z_t$ are known as standardized errors (innovations, shocks, errors, residuals). In models where $\sigma_t^2$ is constant (no GARCH part), $\varepsilon_t$ is proportional to $z_t$ (just their variances differ) and you need not standardize but rather can work directly with $\varepsilon_t$.

You examine ACF and PACF of $\hat z_t\color{blue}{^{**}}$ as the i.i.d. assumption rules out nonzero autocorrelation in $z_t$. You examine ACF and PACF of $\hat z_t^2$ as the i.i.d. assumption rules out autoregressive conditional heteroskedasticity in them.

$\color{blue}{^{*}}$ This is not restrictive but taken as an example to make the discussion concrete.
$\color{blue}{^{**}}$Or $\hat\varepsilon_t$ if $\sigma_t^2=\sigma^2 \ \forall \ t$ here and further. Hats denote fitted values.