I learned how to see quotient groups intuitively when I learned of a group mod its commutator subgroup. If we take a group and mod out all the elements that do not commute, we get a quotient group which is abelian. Simple, nothing technical.
Now I am having trouble seeing a polynomial ring mod some ideal.
Example $1$: Evaluation hom. at $0$: $\ \ \ \varphi: \mathbb Z[x] \to \mathbb Z$ where $\varphi(f(x))=f(0)$. This hom. is surjective and its kernel is the principal ideal generated by $x$, that is $(x)$. So, $\mathbb Z[x]/(x) \cong \mathbb Z$.
Example $2$: Evaluation hom. at $\sqrt 2$: $\ \ \ \psi: \mathbb Q[x] \to \mathbb Q[\sqrt 2]$. We know $\mathbb Q[x]/(x^2-2) \cong \mathbb Q[\sqrt 2]$.
What is $\mathbb Z[x]/(x)$? What do the cosets look like?
What is $\mathbb Q[x]/(x^2-2)$? What do the cosets look like?
Intuitively, what is this telling me? Any other concrete examples for intuition of a quotient ring?