Define the polynomial ring quotient $R = \mathbb{R}[x]/x^n$. Is my understanding correct that
$$ R/x \cong \mathbb{R} $$
and accordingly, as scalars divide all polynomials,
$$R/(R/x) \cong \{1\}$$
If not, how would you characterize $R/x$ and $R/(R/x)$?
I'd prefer to use the common parenthesized notation for ideals, i.e., $R=\Bbb R[x]/(x^n)$ etc.
Strictly speaking, $x\notin R$ and hence $(x)$ is nor n ideal of $R$. Instead of $x$, we should use the image of $x$ (and element of $\Bbb R[x]$) under the canonical projection to $R$, aka. the residue class of $x$, or $x+(x^n)$. With these caveats, yes, we have $$ R/(x+(x^n))\cong \Bbb R$$ per the obvious homomorphism.
However, there is no good way to view $R/(x+(x^n))$ as an ideal of $R$ (even on the level of mere sets, viewing the elements of $R/(x+(x^n))$ as the constant polynomials in $R$ or $\Bbb R[x]$ is not natural), hence we cannot speak of a quotient $R/(R/(x+(x^n)))$.